| L(s) = 1 | − 0.578·2-s − 0.551·3-s − 1.66·4-s + 0.319·6-s − 4.66·7-s + 2.11·8-s − 2.69·9-s − 1.22·11-s + 0.919·12-s − 5.34·13-s + 2.69·14-s + 2.10·16-s + 1.41·17-s + 1.55·18-s + 2.57·21-s + 0.709·22-s + 1.95·23-s − 1.16·24-s + 3.08·26-s + 3.14·27-s + 7.76·28-s − 7.32·29-s + 1.83·31-s − 5.45·32-s + 0.676·33-s − 0.817·34-s + 4.49·36-s + ⋯ |
| L(s) = 1 | − 0.408·2-s − 0.318·3-s − 0.832·4-s + 0.130·6-s − 1.76·7-s + 0.749·8-s − 0.898·9-s − 0.369·11-s + 0.265·12-s − 1.48·13-s + 0.720·14-s + 0.526·16-s + 0.342·17-s + 0.367·18-s + 0.561·21-s + 0.151·22-s + 0.407·23-s − 0.238·24-s + 0.605·26-s + 0.604·27-s + 1.46·28-s − 1.36·29-s + 0.329·31-s − 0.964·32-s + 0.117·33-s − 0.140·34-s + 0.748·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + 0.578T + 2T^{2} \) |
| 3 | \( 1 + 0.551T + 3T^{2} \) |
| 7 | \( 1 + 4.66T + 7T^{2} \) |
| 11 | \( 1 + 1.22T + 11T^{2} \) |
| 13 | \( 1 + 5.34T + 13T^{2} \) |
| 17 | \( 1 - 1.41T + 17T^{2} \) |
| 23 | \( 1 - 1.95T + 23T^{2} \) |
| 29 | \( 1 + 7.32T + 29T^{2} \) |
| 31 | \( 1 - 1.83T + 31T^{2} \) |
| 37 | \( 1 - 5.59T + 37T^{2} \) |
| 41 | \( 1 + 8.29T + 41T^{2} \) |
| 43 | \( 1 - 8.30T + 43T^{2} \) |
| 47 | \( 1 - 4.10T + 47T^{2} \) |
| 53 | \( 1 - 12.9T + 53T^{2} \) |
| 59 | \( 1 + 3.76T + 59T^{2} \) |
| 61 | \( 1 + 4.63T + 61T^{2} \) |
| 67 | \( 1 + 4.65T + 67T^{2} \) |
| 71 | \( 1 - 8.44T + 71T^{2} \) |
| 73 | \( 1 + 4.99T + 73T^{2} \) |
| 79 | \( 1 - 14.7T + 79T^{2} \) |
| 83 | \( 1 + 1.02T + 83T^{2} \) |
| 89 | \( 1 + 3.94T + 89T^{2} \) |
| 97 | \( 1 + 4.72T + 97T^{2} \) |
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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
−7.41030477578475618155266737122, −6.81665113353657595327069989044, −5.86083322087158128835778196400, −5.47812016410771966739728861807, −4.67343480813161172923429921004, −3.78226025834481108015668348826, −3.04387724895260126774375582661, −2.33831166464360547662437562624, −0.69862851590904290930413787088, 0,
0.69862851590904290930413787088, 2.33831166464360547662437562624, 3.04387724895260126774375582661, 3.78226025834481108015668348826, 4.67343480813161172923429921004, 5.47812016410771966739728861807, 5.86083322087158128835778196400, 6.81665113353657595327069989044, 7.41030477578475618155266737122
