| L(s) = 1 | − 3·4-s − 6·7-s + 5·16-s + 2·17-s + 12·19-s + 10·23-s − 4·25-s + 18·28-s + 8·29-s + 6·32-s + 6·37-s − 12·41-s − 2·43-s + 10·49-s + 24·53-s + 12·59-s − 28·61-s − 6·64-s + 6·67-s − 6·68-s − 36·76-s − 16·79-s − 24·89-s − 30·92-s − 30·97-s + 12·100-s + 4·101-s + ⋯ |
| L(s) = 1 | − 3/2·4-s − 2.26·7-s + 5/4·16-s + 0.485·17-s + 2.75·19-s + 2.08·23-s − 4/5·25-s + 3.40·28-s + 1.48·29-s + 1.06·32-s + 0.986·37-s − 1.87·41-s − 0.304·43-s + 10/7·49-s + 3.29·53-s + 1.56·59-s − 3.58·61-s − 3/4·64-s + 0.733·67-s − 0.727·68-s − 4.12·76-s − 1.80·79-s − 2.54·89-s − 3.12·92-s − 3.04·97-s + 6/5·100-s + 0.398·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.045669477\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.045669477\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( ( 1 + T^{2} )^{4} \) |
| 13 | \( 1 + 16 T^{2} + 96 T^{3} + 30 T^{4} + 96 p T^{5} + 16 p^{2} T^{6} + p^{4} T^{8} \) |
| good | 2 | \( ( 1 - p T + p T^{2} - p T^{3} + T^{4} - p^{2} T^{5} + p^{3} T^{6} - p^{4} T^{7} + p^{4} T^{8} )( 1 + p T + 5 T^{2} + p^{3} T^{3} + 13 T^{4} + p^{4} T^{5} + 5 p^{2} T^{6} + p^{4} T^{7} + p^{4} T^{8} ) \) |
| 7 | \( 1 + 6 T + 26 T^{2} + 12 p T^{3} + 229 T^{4} + 60 p T^{5} + 206 T^{6} - 954 T^{7} - 4268 T^{8} - 954 p T^{9} + 206 p^{2} T^{10} + 60 p^{4} T^{11} + 229 p^{4} T^{12} + 12 p^{6} T^{13} + 26 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} \) |
| 11 | \( 1 + 14 T^{2} + 97 T^{4} - 182 p T^{6} - 236 p^{2} T^{8} - 182 p^{3} T^{10} + 97 p^{4} T^{12} + 14 p^{6} T^{14} + p^{8} T^{16} \) |
| 17 | \( ( 1 - 16 T + 128 T^{2} - 688 T^{3} + 3022 T^{4} - 688 p T^{5} + 128 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} )( 1 + 14 T + 50 T^{2} - 220 T^{3} - 2129 T^{4} - 220 p T^{5} + 50 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} ) \) |
| 19 | \( ( 1 - 6 T + 37 T^{2} - 150 T^{3} + 492 T^{4} - 150 p T^{5} + 37 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 23 | \( 1 - 10 T + 2 T^{2} + 108 T^{3} + 1013 T^{4} - 2804 T^{5} - 34298 T^{6} + 4266 p T^{7} + 8492 p T^{8} + 4266 p^{2} T^{9} - 34298 p^{2} T^{10} - 2804 p^{3} T^{11} + 1013 p^{4} T^{12} + 108 p^{5} T^{13} + 2 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} \) |
| 29 | \( 1 - 8 T - 34 T^{2} + 528 T^{3} + 353 T^{4} - 19264 T^{5} + 50686 T^{6} + 248280 T^{7} - 2118188 T^{8} + 248280 p T^{9} + 50686 p^{2} T^{10} - 19264 p^{3} T^{11} + 353 p^{4} T^{12} + 528 p^{5} T^{13} - 34 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \) |
| 31 | \( ( 1 - 92 T^{2} + 3846 T^{4} - 92 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( 1 - 6 T + 110 T^{2} - 588 T^{3} + 6289 T^{4} - 36600 T^{5} + 249746 T^{6} - 1600110 T^{7} + 8587516 T^{8} - 1600110 p T^{9} + 249746 p^{2} T^{10} - 36600 p^{3} T^{11} + 6289 p^{4} T^{12} - 588 p^{5} T^{13} + 110 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) |
| 41 | \( ( 1 + 6 T + 93 T^{2} + 486 T^{3} + 5372 T^{4} + 486 p T^{5} + 93 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 43 | \( 1 + 2 T - 150 T^{2} - 188 T^{3} + 13661 T^{4} + 10620 T^{5} - 862418 T^{6} - 185374 T^{7} + 42096180 T^{8} - 185374 p T^{9} - 862418 p^{2} T^{10} + 10620 p^{3} T^{11} + 13661 p^{4} T^{12} - 188 p^{5} T^{13} - 150 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) |
| 47 | \( 1 - 168 T^{2} + 17500 T^{4} - 1260696 T^{6} + 68079174 T^{8} - 1260696 p^{2} T^{10} + 17500 p^{4} T^{12} - 168 p^{6} T^{14} + p^{8} T^{16} \) |
| 53 | \( ( 1 - 12 T + 248 T^{2} - 36 p T^{3} + 20622 T^{4} - 36 p^{2} T^{5} + 248 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 59 | \( 1 - 12 T + 266 T^{2} - 2616 T^{3} + 37117 T^{4} - 331992 T^{5} + 3532226 T^{6} - 27597948 T^{7} + 240376924 T^{8} - 27597948 p T^{9} + 3532226 p^{2} T^{10} - 331992 p^{3} T^{11} + 37117 p^{4} T^{12} - 2616 p^{5} T^{13} + 266 p^{6} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16} \) |
| 61 | \( 1 + 28 T + 282 T^{2} + 1880 T^{3} + 26477 T^{4} + 344016 T^{5} + 2759698 T^{6} + 21027916 T^{7} + 175399068 T^{8} + 21027916 p T^{9} + 2759698 p^{2} T^{10} + 344016 p^{3} T^{11} + 26477 p^{4} T^{12} + 1880 p^{5} T^{13} + 282 p^{6} T^{14} + 28 p^{7} T^{15} + p^{8} T^{16} \) |
| 67 | \( 1 - 6 T + 166 T^{2} - 924 T^{3} + 11089 T^{4} - 78300 T^{5} + 988546 T^{6} - 7031886 T^{7} + 92397772 T^{8} - 7031886 p T^{9} + 988546 p^{2} T^{10} - 78300 p^{3} T^{11} + 11089 p^{4} T^{12} - 924 p^{5} T^{13} + 166 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) |
| 71 | \( 1 + 66 T^{2} - 5411 T^{4} + 1080 T^{5} - 5238 T^{6} - 2109672 T^{7} + 49128204 T^{8} - 2109672 p T^{9} - 5238 p^{2} T^{10} + 1080 p^{3} T^{11} - 5411 p^{4} T^{12} + 66 p^{6} T^{14} + p^{8} T^{16} \) |
| 73 | \( 1 - 352 T^{2} + 64540 T^{4} - 7783072 T^{6} + 668463622 T^{8} - 7783072 p^{2} T^{10} + 64540 p^{4} T^{12} - 352 p^{6} T^{14} + p^{8} T^{16} \) |
| 79 | \( ( 1 + 8 T + 184 T^{2} + 1256 T^{3} + 21022 T^{4} + 1256 p T^{5} + 184 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 83 | \( 1 - 472 T^{2} + 104380 T^{4} - 14464552 T^{6} + 1407855142 T^{8} - 14464552 p^{2} T^{10} + 104380 p^{4} T^{12} - 472 p^{6} T^{14} + p^{8} T^{16} \) |
| 89 | \( 1 + 24 T + 338 T^{2} + 3504 T^{3} + 22477 T^{4} + 143640 T^{5} + 783482 T^{6} + 3203040 T^{7} + 47062828 T^{8} + 3203040 p T^{9} + 783482 p^{2} T^{10} + 143640 p^{3} T^{11} + 22477 p^{4} T^{12} + 3504 p^{5} T^{13} + 338 p^{6} T^{14} + 24 p^{7} T^{15} + p^{8} T^{16} \) |
| 97 | \( 1 + 30 T + 746 T^{2} + 13380 T^{3} + 217333 T^{4} + 2990232 T^{5} + 37765718 T^{6} + 423036774 T^{7} + 4390187620 T^{8} + 423036774 p T^{9} + 37765718 p^{2} T^{10} + 2990232 p^{3} T^{11} + 217333 p^{4} T^{12} + 13380 p^{5} T^{13} + 746 p^{6} T^{14} + 30 p^{7} T^{15} + p^{8} T^{16} \) |
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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
−4.68675036033394372304599527408, −4.66027143459471635066458444635, −4.49378632444673490798062651282, −4.16340033708136951503235893315, −4.16208190302162688224111251859, −4.07913668123172208420083581093, −3.63627390112111788507055272873, −3.61485854056919118572135320720, −3.50220205801657150572775888764, −3.43010500714002175036193578080, −3.25486033431590743931193426339, −3.13019511639156541836219237192, −3.07043982705124018575311043362, −2.83400809327471299666146453594, −2.55521169258056016509281208109, −2.54575940764480281079593014839, −2.47757698319561562827675623969, −2.04406286236363011702632891925, −1.93934773045573340028326406107, −1.24123091516811397613272351305, −1.23550543909298884765894061055, −1.15885647089443219427386733001, −1.13585218698366801377739690055, −0.50982224875709423994697132729, −0.25691910531959078533441559273,
0.25691910531959078533441559273, 0.50982224875709423994697132729, 1.13585218698366801377739690055, 1.15885647089443219427386733001, 1.23550543909298884765894061055, 1.24123091516811397613272351305, 1.93934773045573340028326406107, 2.04406286236363011702632891925, 2.47757698319561562827675623969, 2.54575940764480281079593014839, 2.55521169258056016509281208109, 2.83400809327471299666146453594, 3.07043982705124018575311043362, 3.13019511639156541836219237192, 3.25486033431590743931193426339, 3.43010500714002175036193578080, 3.50220205801657150572775888764, 3.61485854056919118572135320720, 3.63627390112111788507055272873, 4.07913668123172208420083581093, 4.16208190302162688224111251859, 4.16340033708136951503235893315, 4.49378632444673490798062651282, 4.66027143459471635066458444635, 4.68675036033394372304599527408
Plot not available for L-functions of degree greater than 10.
