| 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 | \( 1 - 4.72T + 97T^{2} \) |
| 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
−7.77318281135443316098921179266, −6.52576482771435321993990644959, −5.84328112175659828498578299747, −5.09782358931374911497851541844, −4.81504666241851513767926048570, −3.73958679087909266117160461496, −3.37845046499347902423541520846, −2.13095226853377100031144017251, −1.38497080976757700598432207782, 0,
1.38497080976757700598432207782, 2.13095226853377100031144017251, 3.37845046499347902423541520846, 3.73958679087909266117160461496, 4.81504666241851513767926048570, 5.09782358931374911497851541844, 5.84328112175659828498578299747, 6.52576482771435321993990644959, 7.77318281135443316098921179266
