| L(s) = 1 | + 0.246·2-s − 0.554·3-s − 1.93·4-s + 5-s − 0.137·6-s + 7-s − 0.972·8-s − 2.69·9-s + 0.246·10-s + 0.603·11-s + 1.07·12-s − 1.64·13-s + 0.246·14-s − 0.554·15-s + 3.63·16-s − 17-s − 0.664·18-s − 6.20·19-s − 1.93·20-s − 0.554·21-s + 0.149·22-s − 5.98·23-s + 0.539·24-s + 25-s − 0.405·26-s + 3.15·27-s − 1.93·28-s + ⋯ |
| L(s) = 1 | + 0.174·2-s − 0.320·3-s − 0.969·4-s + 0.447·5-s − 0.0559·6-s + 0.377·7-s − 0.343·8-s − 0.897·9-s + 0.0781·10-s + 0.182·11-s + 0.310·12-s − 0.455·13-s + 0.0660·14-s − 0.143·15-s + 0.909·16-s − 0.242·17-s − 0.156·18-s − 1.42·19-s − 0.433·20-s − 0.121·21-s + 0.0317·22-s − 1.24·23-s + 0.110·24-s + 0.200·25-s − 0.0795·26-s + 0.607·27-s − 0.366·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 595 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 595 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 2 | \( 1 - 0.246T + 2T^{2} \) |
| 3 | \( 1 + 0.554T + 3T^{2} \) |
| 11 | \( 1 - 0.603T + 11T^{2} \) |
| 13 | \( 1 + 1.64T + 13T^{2} \) |
| 19 | \( 1 + 6.20T + 19T^{2} \) |
| 23 | \( 1 + 5.98T + 23T^{2} \) |
| 29 | \( 1 + 1.86T + 29T^{2} \) |
| 31 | \( 1 + 8.07T + 31T^{2} \) |
| 37 | \( 1 + 10.2T + 37T^{2} \) |
| 41 | \( 1 + 1.86T + 41T^{2} \) |
| 43 | \( 1 - 10.5T + 43T^{2} \) |
| 47 | \( 1 - 2.48T + 47T^{2} \) |
| 53 | \( 1 + 12.2T + 53T^{2} \) |
| 59 | \( 1 + 1.15T + 59T^{2} \) |
| 61 | \( 1 - 3.82T + 61T^{2} \) |
| 67 | \( 1 + 5.37T + 67T^{2} \) |
| 71 | \( 1 - 8.64T + 71T^{2} \) |
| 73 | \( 1 + 7.30T + 73T^{2} \) |
| 79 | \( 1 - 1.34T + 79T^{2} \) |
| 83 | \( 1 - 5.51T + 83T^{2} \) |
| 89 | \( 1 - 15.5T + 89T^{2} \) |
| 97 | \( 1 - 11.8T + 97T^{2} \) |
| 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
−10.30769469061809914947925815986, −9.209592428586304948603158629247, −8.680820067879480429552880861717, −7.70700498545782969008567819272, −6.32832242651816703329953248424, −5.57156548669305740521378301100, −4.70804400867772419094077022501, −3.64734492679080080830549651447, −2.08654824939157025667547261351, 0,
2.08654824939157025667547261351, 3.64734492679080080830549651447, 4.70804400867772419094077022501, 5.57156548669305740521378301100, 6.32832242651816703329953248424, 7.70700498545782969008567819272, 8.680820067879480429552880861717, 9.209592428586304948603158629247, 10.30769469061809914947925815986
