| L(s) = 1 | + 2·2-s + 3.03·3-s + 4·4-s + 6.07·6-s − 1.93·7-s + 8·8-s − 17.7·9-s + 39.0·11-s + 12.1·12-s + 6.52·13-s − 3.87·14-s + 16·16-s + 117.·17-s − 35.5·18-s + 98.9·19-s − 5.88·21-s + 78.1·22-s − 72.0·23-s + 24.2·24-s + 13.0·26-s − 135.·27-s − 7.75·28-s − 36.7·29-s − 178.·31-s + 32·32-s + 118.·33-s + 235.·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.584·3-s + 0.5·4-s + 0.413·6-s − 0.104·7-s + 0.353·8-s − 0.658·9-s + 1.07·11-s + 0.292·12-s + 0.139·13-s − 0.0740·14-s + 0.250·16-s + 1.68·17-s − 0.465·18-s + 1.19·19-s − 0.0611·21-s + 0.757·22-s − 0.653·23-s + 0.206·24-s + 0.0983·26-s − 0.969·27-s − 0.0523·28-s − 0.235·29-s − 1.03·31-s + 0.176·32-s + 0.625·33-s + 1.18·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1250 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1250 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.630392898\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.630392898\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 3.03T + 27T^{2} \) |
| 7 | \( 1 + 1.93T + 343T^{2} \) |
| 11 | \( 1 - 39.0T + 1.33e3T^{2} \) |
| 13 | \( 1 - 6.52T + 2.19e3T^{2} \) |
| 17 | \( 1 - 117.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 98.9T + 6.85e3T^{2} \) |
| 23 | \( 1 + 72.0T + 1.21e4T^{2} \) |
| 29 | \( 1 + 36.7T + 2.43e4T^{2} \) |
| 31 | \( 1 + 178.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 101.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 215.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 276.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 138.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 645.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 490.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 519.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 870.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 777.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 287.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 403.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 593.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.31e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.08e3T + 9.12e5T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.451672406466461490819494276261, −8.390583454944536423609063641356, −7.70516118798755254645589703381, −6.80496543205917213693286388724, −5.79972733406773710335156332759, −5.22532657761834303850279940591, −3.74402232252811139794436543552, −3.43854475122167975676725350986, −2.22847822486751248622569751936, −1.00269062664159924480842669595,
1.00269062664159924480842669595, 2.22847822486751248622569751936, 3.43854475122167975676725350986, 3.74402232252811139794436543552, 5.22532657761834303850279940591, 5.79972733406773710335156332759, 6.80496543205917213693286388724, 7.70516118798755254645589703381, 8.390583454944536423609063641356, 9.451672406466461490819494276261
