| L(s) = 1 | − 8·31-s + 8·43-s + 28·49-s − 8·53-s − 24·67-s − 40·71-s − 16·79-s − 32·83-s − 32·107-s + 56·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 60·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | − 1.43·31-s + 1.21·43-s + 4·49-s − 1.09·53-s − 2.93·67-s − 4.74·71-s − 1.80·79-s − 3.51·83-s − 3.09·107-s + 5.09·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 4.61·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{16} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{16} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(9.170146269\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.170146269\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 4 p T^{2} + 330 T^{4} - 2224 T^{6} + 13203 T^{8} - 2224 p^{2} T^{10} + 330 p^{4} T^{12} - 4 p^{7} T^{14} + p^{8} T^{16} \) |
| 11 | \( 1 - 56 T^{2} + 1612 T^{4} - 29896 T^{6} + 388998 T^{8} - 29896 p^{2} T^{10} + 1612 p^{4} T^{12} - 56 p^{6} T^{14} + p^{8} T^{16} \) |
| 13 | \( ( 1 + 30 T^{2} + 32 T^{3} + 451 T^{4} + 32 p T^{5} + 30 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 17 | \( 1 - 56 T^{2} + 1484 T^{4} - 24968 T^{6} + 374950 T^{8} - 24968 p^{2} T^{10} + 1484 p^{4} T^{12} - 56 p^{6} T^{14} + p^{8} T^{16} \) |
| 19 | \( 1 - 36 T^{2} + 1546 T^{4} - 35120 T^{6} + 832243 T^{8} - 35120 p^{2} T^{10} + 1546 p^{4} T^{12} - 36 p^{6} T^{14} + p^{8} T^{16} \) |
| 23 | \( 1 - 56 T^{2} + 1324 T^{4} - 1592 p T^{6} + 1094310 T^{8} - 1592 p^{3} T^{10} + 1324 p^{4} T^{12} - 56 p^{6} T^{14} + p^{8} T^{16} \) |
| 29 | \( 1 - 88 T^{2} + 4780 T^{4} - 171048 T^{6} + 5385990 T^{8} - 171048 p^{2} T^{10} + 4780 p^{4} T^{12} - 88 p^{6} T^{14} + p^{8} T^{16} \) |
| 31 | \( ( 1 + 4 T + 54 T^{2} + 168 T^{3} + 2099 T^{4} + 168 p T^{5} + 54 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 + 84 T^{2} + 128 T^{3} + 3542 T^{4} + 128 p T^{5} + 84 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 + 100 T^{2} - 56 T^{3} + 126 p T^{4} - 56 p T^{5} + 100 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 - 4 T + 58 T^{2} - 336 T^{3} + 3379 T^{4} - 336 p T^{5} + 58 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 47 | \( 1 - 232 T^{2} + 26076 T^{4} - 1910872 T^{6} + 102863686 T^{8} - 1910872 p^{2} T^{10} + 26076 p^{4} T^{12} - 232 p^{6} T^{14} + p^{8} T^{16} \) |
| 53 | \( ( 1 + 4 T + 92 T^{2} - 44 T^{3} + 3982 T^{4} - 44 p T^{5} + 92 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 59 | \( 1 - 40 T^{2} + 2364 T^{4} - 112984 T^{6} + 20250598 T^{8} - 112984 p^{2} T^{10} + 2364 p^{4} T^{12} - 40 p^{6} T^{14} + p^{8} T^{16} \) |
| 61 | \( 1 - 252 T^{2} + 32650 T^{4} - 2942672 T^{6} + 202734451 T^{8} - 2942672 p^{2} T^{10} + 32650 p^{4} T^{12} - 252 p^{6} T^{14} + p^{8} T^{16} \) |
| 67 | \( ( 1 + 12 T + 154 T^{2} + 1520 T^{3} + 16755 T^{4} + 1520 p T^{5} + 154 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 + 20 T + 380 T^{2} + 4188 T^{3} + 43342 T^{4} + 4188 p T^{5} + 380 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 73 | \( 1 - 184 T^{2} + 15196 T^{4} - 1235336 T^{6} + 104948486 T^{8} - 1235336 p^{2} T^{10} + 15196 p^{4} T^{12} - 184 p^{6} T^{14} + p^{8} T^{16} \) |
| 79 | \( ( 1 + 8 T + 132 T^{2} + 1032 T^{3} + 16454 T^{4} + 1032 p T^{5} + 132 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 83 | \( ( 1 + 16 T + 276 T^{2} + 2576 T^{3} + 30070 T^{4} + 2576 p T^{5} + 276 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 + 132 T^{2} + 64 T^{3} + 18534 T^{4} + 64 p T^{5} + 132 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( 1 - 356 T^{2} + 80362 T^{4} - 11927760 T^{6} + 1353370099 T^{8} - 11927760 p^{2} T^{10} + 80362 p^{4} T^{12} - 356 p^{6} T^{14} + p^{8} T^{16} \) |
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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
−3.16349545222799665656240172664, −3.13406671654597833591655151063, −3.03826677997511195327725720735, −3.00059583323099788222436953577, −2.59627041256326755561328545844, −2.58279383912367293523651734671, −2.58250807395200471031542364254, −2.55365332866261944014898109291, −2.50823945915714711854140080036, −2.21384963269343973797909212068, −2.16242197821159219763576785387, −1.93427679366342370101674818625, −1.71841357078738863048119860469, −1.71230058828455677105104471050, −1.68670604803841406389663256268, −1.49068784131162374569626402318, −1.28996049907683527003580737242, −1.24127148703789770897009867670, −1.15110248650026510678433915877, −1.14732122101178269342864263724, −0.792345871555447648140673436483, −0.42208753724759524757083428072, −0.34133012771404888550808650973, −0.32568516827759894899725355060, −0.32191194017720178822772274047,
0.32191194017720178822772274047, 0.32568516827759894899725355060, 0.34133012771404888550808650973, 0.42208753724759524757083428072, 0.792345871555447648140673436483, 1.14732122101178269342864263724, 1.15110248650026510678433915877, 1.24127148703789770897009867670, 1.28996049907683527003580737242, 1.49068784131162374569626402318, 1.68670604803841406389663256268, 1.71230058828455677105104471050, 1.71841357078738863048119860469, 1.93427679366342370101674818625, 2.16242197821159219763576785387, 2.21384963269343973797909212068, 2.50823945915714711854140080036, 2.55365332866261944014898109291, 2.58250807395200471031542364254, 2.58279383912367293523651734671, 2.59627041256326755561328545844, 3.00059583323099788222436953577, 3.03826677997511195327725720735, 3.13406671654597833591655151063, 3.16349545222799665656240172664
Plot not available for L-functions of degree greater than 10.
