| L(s) = 1 | − 128·2-s + 1.63e4·4-s + 1.14e5·5-s − 3.03e6·7-s − 2.09e6·8-s − 1.46e7·10-s + 1.03e8·11-s − 1.04e8·13-s + 3.88e8·14-s + 2.68e8·16-s − 9.97e8·17-s + 4.93e9·19-s + 1.88e9·20-s − 1.32e10·22-s − 8.32e9·23-s − 1.73e10·25-s + 1.33e10·26-s − 4.97e10·28-s − 1.04e11·29-s − 2.96e11·31-s − 3.43e10·32-s + 1.27e11·34-s − 3.48e11·35-s − 1.78e11·37-s − 6.31e11·38-s − 2.40e11·40-s + 1.79e12·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.657·5-s − 1.39·7-s − 0.353·8-s − 0.464·10-s + 1.60·11-s − 0.461·13-s + 0.984·14-s + 1/4·16-s − 0.589·17-s + 1.26·19-s + 0.328·20-s − 1.13·22-s − 0.509·23-s − 0.568·25-s + 0.326·26-s − 0.696·28-s − 1.12·29-s − 1.93·31-s − 0.176·32-s + 0.416·34-s − 0.915·35-s − 0.308·37-s − 0.895·38-s − 0.232·40-s + 1.43·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(8)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{17}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p^{7} T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 22962 p T + p^{15} T^{2} \) |
| 7 | \( 1 + 433504 p T + p^{15} T^{2} \) |
| 11 | \( 1 - 9404700 p T + p^{15} T^{2} \) |
| 13 | \( 1 + 104365834 T + p^{15} T^{2} \) |
| 17 | \( 1 + 997689762 T + p^{15} T^{2} \) |
| 19 | \( 1 - 4934015444 T + p^{15} T^{2} \) |
| 23 | \( 1 + 8324920200 T + p^{15} T^{2} \) |
| 29 | \( 1 + 104128242846 T + p^{15} T^{2} \) |
| 31 | \( 1 + 296696681512 T + p^{15} T^{2} \) |
| 37 | \( 1 + 178337455666 T + p^{15} T^{2} \) |
| 41 | \( 1 - 1790882416086 T + p^{15} T^{2} \) |
| 43 | \( 1 + 2863459422772 T + p^{15} T^{2} \) |
| 47 | \( 1 + 4332907521600 T + p^{15} T^{2} \) |
| 53 | \( 1 + 9732317104422 T + p^{15} T^{2} \) |
| 59 | \( 1 - 13514837176500 T + p^{15} T^{2} \) |
| 61 | \( 1 - 5352663511190 T + p^{15} T^{2} \) |
| 67 | \( 1 + 53233909720108 T + p^{15} T^{2} \) |
| 71 | \( 1 - 20229661643400 T + p^{15} T^{2} \) |
| 73 | \( 1 - 26264166466106 T + p^{15} T^{2} \) |
| 79 | \( 1 + 339031361615128 T + p^{15} T^{2} \) |
| 83 | \( 1 + 131684771045076 T + p^{15} T^{2} \) |
| 89 | \( 1 - 39352148322678 T + p^{15} T^{2} \) |
| 97 | \( 1 - 1128750908801474 T + p^{15} T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.44608424228158742869394723632, −12.98983876133781325923922177900, −11.56502509819906390217605032945, −9.769425179431190076795337099692, −9.232833537558887637727500075404, −7.10612677814630602534557530191, −5.96793062290413971209930034306, −3.48802857473852271989012553066, −1.72362200652957003888212208484, 0,
1.72362200652957003888212208484, 3.48802857473852271989012553066, 5.96793062290413971209930034306, 7.10612677814630602534557530191, 9.232833537558887637727500075404, 9.769425179431190076795337099692, 11.56502509819906390217605032945, 12.98983876133781325923922177900, 14.44608424228158742869394723632
