| L(s) = 1 | + 5·2-s − 3·3-s + 17·4-s + 5·5-s − 15·6-s + 7·7-s + 45·8-s + 9·9-s + 25·10-s + 12·11-s − 51·12-s + 30·13-s + 35·14-s − 15·15-s + 89·16-s − 134·17-s + 45·18-s − 92·19-s + 85·20-s − 21·21-s + 60·22-s + 112·23-s − 135·24-s + 25·25-s + 150·26-s − 27·27-s + 119·28-s + ⋯ |
| L(s) = 1 | + 1.76·2-s − 0.577·3-s + 17/8·4-s + 0.447·5-s − 1.02·6-s + 0.377·7-s + 1.98·8-s + 1/3·9-s + 0.790·10-s + 0.328·11-s − 1.22·12-s + 0.640·13-s + 0.668·14-s − 0.258·15-s + 1.39·16-s − 1.91·17-s + 0.589·18-s − 1.11·19-s + 0.950·20-s − 0.218·21-s + 0.581·22-s + 1.01·23-s − 1.14·24-s + 1/5·25-s + 1.13·26-s − 0.192·27-s + 0.803·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.783803180\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.783803180\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + p T \) |
| 5 | \( 1 - p T \) |
| 7 | \( 1 - p T \) |
| good | 2 | \( 1 - 5 T + p^{3} T^{2} \) |
| 11 | \( 1 - 12 T + p^{3} T^{2} \) |
| 13 | \( 1 - 30 T + p^{3} T^{2} \) |
| 17 | \( 1 + 134 T + p^{3} T^{2} \) |
| 19 | \( 1 + 92 T + p^{3} T^{2} \) |
| 23 | \( 1 - 112 T + p^{3} T^{2} \) |
| 29 | \( 1 + 2 p T + p^{3} T^{2} \) |
| 31 | \( 1 + 224 T + p^{3} T^{2} \) |
| 37 | \( 1 + 146 T + p^{3} T^{2} \) |
| 41 | \( 1 - 18 T + p^{3} T^{2} \) |
| 43 | \( 1 - 340 T + p^{3} T^{2} \) |
| 47 | \( 1 - 208 T + p^{3} T^{2} \) |
| 53 | \( 1 + 754 T + p^{3} T^{2} \) |
| 59 | \( 1 - 380 T + p^{3} T^{2} \) |
| 61 | \( 1 - 718 T + p^{3} T^{2} \) |
| 67 | \( 1 - 412 T + p^{3} T^{2} \) |
| 71 | \( 1 + 960 T + p^{3} T^{2} \) |
| 73 | \( 1 - 1066 T + p^{3} T^{2} \) |
| 79 | \( 1 - 896 T + p^{3} T^{2} \) |
| 83 | \( 1 - 436 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1038 T + p^{3} T^{2} \) |
| 97 | \( 1 + 702 T + p^{3} T^{2} \) |
| 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
−13.16198534108093503000155853587, −12.62064443947284468305262868757, −11.19171774196022368045326514221, −10.91806218334070416481062719828, −8.937392506602025682258189310005, −6.97138698939960924788133947488, −6.15891394011823901007623239547, −5.01733452429139859066853582881, −3.96372846132159525231642033482, −2.09184259546574420385600615534,
2.09184259546574420385600615534, 3.96372846132159525231642033482, 5.01733452429139859066853582881, 6.15891394011823901007623239547, 6.97138698939960924788133947488, 8.937392506602025682258189310005, 10.91806218334070416481062719828, 11.19171774196022368045326514221, 12.62064443947284468305262868757, 13.16198534108093503000155853587
