| L(s) = 1 | − 1.31·5-s − 2.21·7-s + 3.68·11-s + 1.59·13-s − 7.64·17-s + 7.39·19-s + 7.73·23-s − 3.28·25-s + 7.92·29-s − 2.95·31-s + 2.90·35-s − 5.70·37-s − 41-s − 6.05·43-s − 5.64·47-s − 2.09·49-s + 2.14·53-s − 4.83·55-s − 7.47·59-s − 4.67·61-s − 2.08·65-s + 11.9·67-s + 2.97·71-s − 1.19·73-s − 8.16·77-s + 5.67·79-s + 2.92·83-s + ⋯ |
| L(s) = 1 | − 0.586·5-s − 0.836·7-s + 1.11·11-s + 0.441·13-s − 1.85·17-s + 1.69·19-s + 1.61·23-s − 0.656·25-s + 1.47·29-s − 0.530·31-s + 0.490·35-s − 0.938·37-s − 0.156·41-s − 0.922·43-s − 0.823·47-s − 0.299·49-s + 0.295·53-s − 0.652·55-s − 0.973·59-s − 0.598·61-s − 0.258·65-s + 1.45·67-s + 0.353·71-s − 0.139·73-s − 0.930·77-s + 0.638·79-s + 0.320·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5904 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5904 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.515448086\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.515448086\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 5 | \( 1 + 1.31T + 5T^{2} \) |
| 7 | \( 1 + 2.21T + 7T^{2} \) |
| 11 | \( 1 - 3.68T + 11T^{2} \) |
| 13 | \( 1 - 1.59T + 13T^{2} \) |
| 17 | \( 1 + 7.64T + 17T^{2} \) |
| 19 | \( 1 - 7.39T + 19T^{2} \) |
| 23 | \( 1 - 7.73T + 23T^{2} \) |
| 29 | \( 1 - 7.92T + 29T^{2} \) |
| 31 | \( 1 + 2.95T + 31T^{2} \) |
| 37 | \( 1 + 5.70T + 37T^{2} \) |
| 43 | \( 1 + 6.05T + 43T^{2} \) |
| 47 | \( 1 + 5.64T + 47T^{2} \) |
| 53 | \( 1 - 2.14T + 53T^{2} \) |
| 59 | \( 1 + 7.47T + 59T^{2} \) |
| 61 | \( 1 + 4.67T + 61T^{2} \) |
| 67 | \( 1 - 11.9T + 67T^{2} \) |
| 71 | \( 1 - 2.97T + 71T^{2} \) |
| 73 | \( 1 + 1.19T + 73T^{2} \) |
| 79 | \( 1 - 5.67T + 79T^{2} \) |
| 83 | \( 1 - 2.92T + 83T^{2} \) |
| 89 | \( 1 + 6.85T + 89T^{2} \) |
| 97 | \( 1 - 3.07T + 97T^{2} \) |
| 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
−8.190190909919361473937461175373, −7.12928515140730549217471399666, −6.79004311360701118954929818068, −6.17169077535249461975594925425, −5.09979861280525264281372400891, −4.43650799940488520456133114182, −3.48463137966782997359392773925, −3.10578996004925020851417815469, −1.77108400540176279476242777012, −0.65700592045238220942559790668,
0.65700592045238220942559790668, 1.77108400540176279476242777012, 3.10578996004925020851417815469, 3.48463137966782997359392773925, 4.43650799940488520456133114182, 5.09979861280525264281372400891, 6.17169077535249461975594925425, 6.79004311360701118954929818068, 7.12928515140730549217471399666, 8.190190909919361473937461175373
