| L(s) = 1 | − 2-s + 4-s + 5-s + 7-s − 8-s − 10-s + 2·13-s − 14-s + 16-s + 17-s − 4·19-s + 20-s + 25-s − 2·26-s + 28-s − 6·29-s + 8·31-s − 32-s − 34-s + 35-s + 2·37-s + 4·38-s − 40-s − 6·41-s − 4·43-s − 12·47-s + 49-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.377·7-s − 0.353·8-s − 0.316·10-s + 0.554·13-s − 0.267·14-s + 1/4·16-s + 0.242·17-s − 0.917·19-s + 0.223·20-s + 1/5·25-s − 0.392·26-s + 0.188·28-s − 1.11·29-s + 1.43·31-s − 0.176·32-s − 0.171·34-s + 0.169·35-s + 0.328·37-s + 0.648·38-s − 0.158·40-s − 0.937·41-s − 0.609·43-s − 1.75·47-s + 1/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10710 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
| 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
−16.97931368584263, −16.41820876856211, −15.70516369782868, −15.19086645928072, −14.62446322968118, −14.04958301496418, −13.26712575891940, −12.93445912519119, −12.03391476296415, −11.54421269729425, −10.91136744072833, −10.38792362437440, −9.778887791804477, −9.208857765924295, −8.479895938198790, −8.088111019991270, −7.386757828468904, −6.426465118361218, −6.253528560414446, −5.257563740393668, −4.602785458451671, −3.642942152715429, −2.846090608724181, −1.905005984060601, −1.297361900472208, 0,
1.297361900472208, 1.905005984060601, 2.846090608724181, 3.642942152715429, 4.602785458451671, 5.257563740393668, 6.253528560414446, 6.426465118361218, 7.386757828468904, 8.088111019991270, 8.479895938198790, 9.208857765924295, 9.778887791804477, 10.38792362437440, 10.91136744072833, 11.54421269729425, 12.03391476296415, 12.93445912519119, 13.26712575891940, 14.04958301496418, 14.62446322968118, 15.19086645928072, 15.70516369782868, 16.41820876856211, 16.97931368584263
