| L(s) = 1 | + 4.53·2-s + 12.5·4-s + 2.63·7-s + 20.6·8-s − 20.9·11-s − 60.9·13-s + 11.9·14-s − 6.70·16-s + 86.8·17-s + 41.8·19-s − 94.9·22-s − 97.3·23-s − 276.·26-s + 33.1·28-s − 157.·29-s + 95.3·31-s − 195.·32-s + 393.·34-s + 160.·37-s + 189.·38-s + 233.·41-s − 487.·43-s − 262.·44-s − 441.·46-s − 24.3·47-s − 336.·49-s − 765.·52-s + ⋯ |
| L(s) = 1 | + 1.60·2-s + 1.57·4-s + 0.142·7-s + 0.914·8-s − 0.573·11-s − 1.30·13-s + 0.228·14-s − 0.104·16-s + 1.23·17-s + 0.504·19-s − 0.919·22-s − 0.882·23-s − 2.08·26-s + 0.223·28-s − 1.00·29-s + 0.552·31-s − 1.08·32-s + 1.98·34-s + 0.714·37-s + 0.809·38-s + 0.888·41-s − 1.72·43-s − 0.900·44-s − 1.41·46-s − 0.0754·47-s − 0.979·49-s − 2.04·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 4.53T + 8T^{2} \) |
| 7 | \( 1 - 2.63T + 343T^{2} \) |
| 11 | \( 1 + 20.9T + 1.33e3T^{2} \) |
| 13 | \( 1 + 60.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 86.8T + 4.91e3T^{2} \) |
| 19 | \( 1 - 41.8T + 6.85e3T^{2} \) |
| 23 | \( 1 + 97.3T + 1.21e4T^{2} \) |
| 29 | \( 1 + 157.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 95.3T + 2.97e4T^{2} \) |
| 37 | \( 1 - 160.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 233.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 487.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 24.3T + 1.03e5T^{2} \) |
| 53 | \( 1 + 709.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 191.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 744.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 823.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.06e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 132.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 704.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.41e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 401.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 530.T + 9.12e5T^{2} \) |
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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
−7.951416817403752567337103708025, −7.58153869707324303838609785484, −6.54706484581487110561854183811, −5.75017694024798114949177005546, −5.08615355579928204817585684446, −4.47553636034340737750829887754, −3.40842711842331990487378409923, −2.75760321447431663318350741149, −1.71166081469413268257127999227, 0,
1.71166081469413268257127999227, 2.75760321447431663318350741149, 3.40842711842331990487378409923, 4.47553636034340737750829887754, 5.08615355579928204817585684446, 5.75017694024798114949177005546, 6.54706484581487110561854183811, 7.58153869707324303838609785484, 7.951416817403752567337103708025
