L(s) = 1 | − 12·3-s + 4·5-s + 24·7-s + 78·9-s + 8·11-s + 24·13-s − 48·15-s − 20·17-s − 24·19-s − 288·21-s − 40·23-s + 14·25-s − 360·27-s − 48·29-s − 96·33-s + 96·35-s − 288·39-s + 60·41-s + 116·43-s + 312·45-s + 68·47-s + 32·49-s + 240·51-s − 168·53-s + 32·55-s + 288·57-s + 156·59-s + ⋯ |
L(s) = 1 | − 4·3-s + 4/5·5-s + 24/7·7-s + 26/3·9-s + 8/11·11-s + 1.84·13-s − 3.19·15-s − 1.17·17-s − 1.26·19-s − 13.7·21-s − 1.73·23-s + 0.559·25-s − 13.3·27-s − 1.65·29-s − 2.90·33-s + 2.74·35-s − 7.38·39-s + 1.46·41-s + 2.69·43-s + 6.93·45-s + 1.44·47-s + 0.653·49-s + 4.70·51-s − 3.16·53-s + 0.581·55-s + 5.05·57-s + 2.64·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.1845886433\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1845886433\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( ( 1 + p T + p T^{2} )^{4} \) |
| 19 | \( 1 + 24 T + 80 T^{2} - 312 p T^{3} - 270 p^{2} T^{4} - 312 p^{3} T^{5} + 80 p^{4} T^{6} + 24 p^{6} T^{7} + p^{8} T^{8} \) |
good | 5 | \( 1 - 4 T + 2 T^{2} + 232 T^{3} - 1588 T^{4} + 4048 T^{5} + 6924 T^{6} - 170652 T^{7} + 927739 T^{8} - 170652 p^{2} T^{9} + 6924 p^{4} T^{10} + 4048 p^{6} T^{11} - 1588 p^{8} T^{12} + 232 p^{10} T^{13} + 2 p^{12} T^{14} - 4 p^{14} T^{15} + p^{16} T^{16} \) |
| 7 | \( ( 1 - 12 T + 200 T^{2} - 240 p T^{3} + 14703 T^{4} - 240 p^{3} T^{5} + 200 p^{4} T^{6} - 12 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 11 | \( ( 1 - 4 T + 254 T^{2} - 1512 T^{3} + 34088 T^{4} - 1512 p^{2} T^{5} + 254 p^{4} T^{6} - 4 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 13 | \( 1 - 24 T + 538 T^{2} - 8304 T^{3} + 94729 T^{4} - 127440 T^{5} - 9926390 T^{6} + 330876120 T^{7} - 5113852844 T^{8} + 330876120 p^{2} T^{9} - 9926390 p^{4} T^{10} - 127440 p^{6} T^{11} + 94729 p^{8} T^{12} - 8304 p^{10} T^{13} + 538 p^{12} T^{14} - 24 p^{14} T^{15} + p^{16} T^{16} \) |
| 17 | \( 1 + 20 T + 176 T^{2} - 7064 T^{3} - 13766 p T^{4} - 3789284 T^{5} - 430560 T^{6} + 1083684252 T^{7} + 26994491251 T^{8} + 1083684252 p^{2} T^{9} - 430560 p^{4} T^{10} - 3789284 p^{6} T^{11} - 13766 p^{9} T^{12} - 7064 p^{10} T^{13} + 176 p^{12} T^{14} + 20 p^{14} T^{15} + p^{16} T^{16} \) |
| 23 | \( 1 + 40 T - 850 T^{2} - 26968 T^{3} + 1472036 T^{4} + 23651516 T^{5} - 1022895084 T^{6} - 1707420504 T^{7} + 775693330411 T^{8} - 1707420504 p^{2} T^{9} - 1022895084 p^{4} T^{10} + 23651516 p^{6} T^{11} + 1472036 p^{8} T^{12} - 26968 p^{10} T^{13} - 850 p^{12} T^{14} + 40 p^{14} T^{15} + p^{16} T^{16} \) |
| 29 | \( 1 + 48 T + 4108 T^{2} + 160320 T^{3} + 8781946 T^{4} + 283206672 T^{5} + 12015752560 T^{6} + 332550420528 T^{7} + 11826496442035 T^{8} + 332550420528 p^{2} T^{9} + 12015752560 p^{4} T^{10} + 283206672 p^{6} T^{11} + 8781946 p^{8} T^{12} + 160320 p^{10} T^{13} + 4108 p^{12} T^{14} + 48 p^{14} T^{15} + p^{16} T^{16} \) |
| 31 | \( 1 - 4400 T^{2} + 10278382 T^{4} - 15849053696 T^{6} + 17739136481587 T^{8} - 15849053696 p^{4} T^{10} + 10278382 p^{8} T^{12} - 4400 p^{12} T^{14} + p^{16} T^{16} \) |
| 37 | \( 1 - 7868 T^{2} + 29439082 T^{4} - 69087044720 T^{6} + 112166334644467 T^{8} - 69087044720 p^{4} T^{10} + 29439082 p^{8} T^{12} - 7868 p^{12} T^{14} + p^{16} T^{16} \) |
| 41 | \( 1 - 60 T + 5272 T^{2} - 244320 T^{3} + 10680970 T^{4} - 12481020 p T^{5} + 26515297024 T^{6} - 29352654300 p T^{7} + 63071084939251 T^{8} - 29352654300 p^{3} T^{9} + 26515297024 p^{4} T^{10} - 12481020 p^{7} T^{11} + 10680970 p^{8} T^{12} - 244320 p^{10} T^{13} + 5272 p^{12} T^{14} - 60 p^{14} T^{15} + p^{16} T^{16} \) |
| 43 | \( 1 - 116 T + 6864 T^{2} - 445312 T^{3} + 21426761 T^{4} - 534739944 T^{5} + 13218576448 T^{6} - 14635797716 T^{7} - 24991310356848 T^{8} - 14635797716 p^{2} T^{9} + 13218576448 p^{4} T^{10} - 534739944 p^{6} T^{11} + 21426761 p^{8} T^{12} - 445312 p^{10} T^{13} + 6864 p^{12} T^{14} - 116 p^{14} T^{15} + p^{16} T^{16} \) |
| 47 | \( 1 - 68 T + 272 T^{2} - 27784 T^{3} + 1244762 T^{4} + 286931828 T^{5} + 2977950240 T^{6} - 749262874476 T^{7} + 17326035283699 T^{8} - 749262874476 p^{2} T^{9} + 2977950240 p^{4} T^{10} + 286931828 p^{6} T^{11} + 1244762 p^{8} T^{12} - 27784 p^{10} T^{13} + 272 p^{12} T^{14} - 68 p^{14} T^{15} + p^{16} T^{16} \) |
| 53 | \( 1 + 168 T + 22078 T^{2} + 2128560 T^{3} + 179216380 T^{4} + 12871089972 T^{5} + 851960139460 T^{6} + 50495390324232 T^{7} + 2809342040388811 T^{8} + 50495390324232 p^{2} T^{9} + 851960139460 p^{4} T^{10} + 12871089972 p^{6} T^{11} + 179216380 p^{8} T^{12} + 2128560 p^{10} T^{13} + 22078 p^{12} T^{14} + 168 p^{14} T^{15} + p^{16} T^{16} \) |
| 59 | \( 1 - 156 T + 15370 T^{2} - 1132248 T^{3} + 63250276 T^{4} - 3313637640 T^{5} + 176247520108 T^{6} - 10449104507436 T^{7} + 652032168648475 T^{8} - 10449104507436 p^{2} T^{9} + 176247520108 p^{4} T^{10} - 3313637640 p^{6} T^{11} + 63250276 p^{8} T^{12} - 1132248 p^{10} T^{13} + 15370 p^{12} T^{14} - 156 p^{14} T^{15} + p^{16} T^{16} \) |
| 61 | \( 1 - 72 T - 9686 T^{2} + 356976 T^{3} + 88286281 T^{4} - 1543556736 T^{5} - 480900464102 T^{6} + 2328443685144 T^{7} + 2016869892781972 T^{8} + 2328443685144 p^{2} T^{9} - 480900464102 p^{4} T^{10} - 1543556736 p^{6} T^{11} + 88286281 p^{8} T^{12} + 356976 p^{10} T^{13} - 9686 p^{12} T^{14} - 72 p^{14} T^{15} + p^{16} T^{16} \) |
| 67 | \( 1 - 108 T + 16624 T^{2} - 1375488 T^{3} + 130111681 T^{4} - 9758246256 T^{5} + 713448829312 T^{6} - 50890778526564 T^{7} + 3295618434822400 T^{8} - 50890778526564 p^{2} T^{9} + 713448829312 p^{4} T^{10} - 9758246256 p^{6} T^{11} + 130111681 p^{8} T^{12} - 1375488 p^{10} T^{13} + 16624 p^{12} T^{14} - 108 p^{14} T^{15} + p^{16} T^{16} \) |
| 71 | \( 1 + 444 T + 109072 T^{2} + 19251840 T^{3} + 2704579930 T^{4} + 316758495564 T^{5} + 31704202831264 T^{6} + 2751602215631484 T^{7} + 208709603420760691 T^{8} + 2751602215631484 p^{2} T^{9} + 31704202831264 p^{4} T^{10} + 316758495564 p^{6} T^{11} + 2704579930 p^{8} T^{12} + 19251840 p^{10} T^{13} + 109072 p^{12} T^{14} + 444 p^{14} T^{15} + p^{16} T^{16} \) |
| 73 | \( 1 + 68 T - 1938 T^{2} - 195128 T^{3} - 29714455 T^{4} - 3118619256 T^{5} + 24870519598 T^{6} + 16879587836012 T^{7} + 1420069594452660 T^{8} + 16879587836012 p^{2} T^{9} + 24870519598 p^{4} T^{10} - 3118619256 p^{6} T^{11} - 29714455 p^{8} T^{12} - 195128 p^{10} T^{13} - 1938 p^{12} T^{14} + 68 p^{14} T^{15} + p^{16} T^{16} \) |
| 79 | \( 1 + 420 T + 103000 T^{2} + 18564000 T^{3} + 2720500513 T^{4} + 338037041880 T^{5} + 36421672129000 T^{6} + 3448357103704020 T^{7} + 289070560542575008 T^{8} + 3448357103704020 p^{2} T^{9} + 36421672129000 p^{4} T^{10} + 338037041880 p^{6} T^{11} + 2720500513 p^{8} T^{12} + 18564000 p^{10} T^{13} + 103000 p^{12} T^{14} + 420 p^{14} T^{15} + p^{16} T^{16} \) |
| 83 | \( ( 1 + 212 T + 40808 T^{2} + 4608060 T^{3} + 466595750 T^{4} + 4608060 p^{2} T^{5} + 40808 p^{4} T^{6} + 212 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 89 | \( 1 + 420 T + 95986 T^{2} + 15618120 T^{3} + 2032524964 T^{4} + 236593477800 T^{5} + 26082584100700 T^{6} + 2708951100422100 T^{7} + 255855910860494107 T^{8} + 2708951100422100 p^{2} T^{9} + 26082584100700 p^{4} T^{10} + 236593477800 p^{6} T^{11} + 2032524964 p^{8} T^{12} + 15618120 p^{10} T^{13} + 95986 p^{12} T^{14} + 420 p^{14} T^{15} + p^{16} T^{16} \) |
| 97 | \( 1 - 156 T + 44500 T^{2} - 5676528 T^{3} + 1036648150 T^{4} - 114563086020 T^{5} + 15984362834656 T^{6} - 1525740713757228 T^{7} + 176299774873016995 T^{8} - 1525740713757228 p^{2} T^{9} + 15984362834656 p^{4} T^{10} - 114563086020 p^{6} T^{11} + 1036648150 p^{8} T^{12} - 5676528 p^{10} T^{13} + 44500 p^{12} T^{14} - 156 p^{14} T^{15} + p^{16} T^{16} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.28596175129995189106771268287, −4.24716681643804907380507317057, −4.05099472073517771901187802616, −3.91076098287910909587854658723, −3.82192244519972237664777908020, −3.77295354778744191538582473122, −3.24049835057699632272606592484, −3.06245901510593223851176809612, −2.95098096261169322514206454310, −2.83123377441316455458644403897, −2.80263036672585474752447118961, −2.37636447006661166785820352780, −2.29356575555278049449524736632, −2.17676295561479527942066427660, −1.77081597085415917246229203745, −1.60830141525183993058712742366, −1.52533237150880618000408715830, −1.47391731319901335738071129213, −1.46134574246074604969637822501, −1.45699781972027698465739487078, −1.10549134334923535469236117714, −0.860333751985016733787490310128, −0.50261317715467303617706866611, −0.14612279639245242176576496109, −0.13011461871277664679276983451,
0.13011461871277664679276983451, 0.14612279639245242176576496109, 0.50261317715467303617706866611, 0.860333751985016733787490310128, 1.10549134334923535469236117714, 1.45699781972027698465739487078, 1.46134574246074604969637822501, 1.47391731319901335738071129213, 1.52533237150880618000408715830, 1.60830141525183993058712742366, 1.77081597085415917246229203745, 2.17676295561479527942066427660, 2.29356575555278049449524736632, 2.37636447006661166785820352780, 2.80263036672585474752447118961, 2.83123377441316455458644403897, 2.95098096261169322514206454310, 3.06245901510593223851176809612, 3.24049835057699632272606592484, 3.77295354778744191538582473122, 3.82192244519972237664777908020, 3.91076098287910909587854658723, 4.05099472073517771901187802616, 4.24716681643804907380507317057, 4.28596175129995189106771268287
Plot not available for L-functions of degree greater than 10.