| L(s) = 1 | − 2.87·3-s − 1.16·7-s − 18.7·9-s + 19.3·11-s + 56.2·13-s − 94.7·17-s − 16.4·19-s + 3.33·21-s − 32.2·23-s + 131.·27-s + 226.·29-s − 312.·31-s − 55.5·33-s + 44.1·37-s − 161.·39-s − 201.·41-s + 384.·43-s − 130.·47-s − 341.·49-s + 272.·51-s + 61.7·53-s + 47.1·57-s + 78.9·59-s − 393.·61-s + 21.7·63-s + 549.·67-s + 92.8·69-s + ⋯ |
| L(s) = 1 | − 0.553·3-s − 0.0626·7-s − 0.693·9-s + 0.529·11-s + 1.20·13-s − 1.35·17-s − 0.198·19-s + 0.0346·21-s − 0.292·23-s + 0.937·27-s + 1.44·29-s − 1.80·31-s − 0.292·33-s + 0.195·37-s − 0.664·39-s − 0.766·41-s + 1.36·43-s − 0.404·47-s − 0.996·49-s + 0.747·51-s + 0.160·53-s + 0.109·57-s + 0.174·59-s − 0.826·61-s + 0.0435·63-s + 1.00·67-s + 0.162·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.351314277\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.351314277\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 2.87T + 27T^{2} \) |
| 7 | \( 1 + 1.16T + 343T^{2} \) |
| 11 | \( 1 - 19.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 56.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + 94.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + 16.4T + 6.85e3T^{2} \) |
| 23 | \( 1 + 32.2T + 1.21e4T^{2} \) |
| 29 | \( 1 - 226.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 312.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 44.1T + 5.06e4T^{2} \) |
| 41 | \( 1 + 201.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 384.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 130.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 61.7T + 1.48e5T^{2} \) |
| 59 | \( 1 - 78.9T + 2.05e5T^{2} \) |
| 61 | \( 1 + 393.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 549.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 348.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 780.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 542.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.10e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 582.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 757.T + 9.12e5T^{2} \) |
| 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
−9.470869972454094234969032454614, −8.774590366917042450750041613684, −8.086931099311821665473754759390, −6.72770816789750133952850693544, −6.29351580567104292286067801114, −5.35137637060830211607175248839, −4.31605661338748667837056359373, −3.32664209859567740182888811998, −2.00398036816840519579443373380, −0.63038696067226468191690049685,
0.63038696067226468191690049685, 2.00398036816840519579443373380, 3.32664209859567740182888811998, 4.31605661338748667837056359373, 5.35137637060830211607175248839, 6.29351580567104292286067801114, 6.72770816789750133952850693544, 8.086931099311821665473754759390, 8.774590366917042450750041613684, 9.470869972454094234969032454614
