| L(s) = 1 | + 28.1·2-s + 281.·4-s − 762.·5-s + 5.57e3·7-s − 6.49e3·8-s − 2.14e4·10-s + 4.76e4·11-s − 9.22e4·13-s + 1.56e5·14-s − 3.27e5·16-s + 8.35e4·17-s − 8.37e3·19-s − 2.14e5·20-s + 1.34e6·22-s − 3.64e5·23-s − 1.37e6·25-s − 2.59e6·26-s + 1.56e6·28-s + 3.50e6·29-s − 5.20e6·31-s − 5.88e6·32-s + 2.35e6·34-s − 4.25e6·35-s − 4.99e5·37-s − 2.35e5·38-s + 4.95e6·40-s + 5.43e6·41-s + ⋯ |
| L(s) = 1 | + 1.24·2-s + 0.549·4-s − 0.545·5-s + 0.877·7-s − 0.560·8-s − 0.679·10-s + 0.981·11-s − 0.895·13-s + 1.09·14-s − 1.24·16-s + 0.242·17-s − 0.0147·19-s − 0.299·20-s + 1.22·22-s − 0.271·23-s − 0.702·25-s − 1.11·26-s + 0.481·28-s + 0.920·29-s − 1.01·31-s − 0.991·32-s + 0.301·34-s − 0.478·35-s − 0.0438·37-s − 0.0183·38-s + 0.306·40-s + 0.300·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 17 | \( 1 - 8.35e4T \) |
| good | 2 | \( 1 - 28.1T + 512T^{2} \) |
| 5 | \( 1 + 762.T + 1.95e6T^{2} \) |
| 7 | \( 1 - 5.57e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 4.76e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 9.22e4T + 1.06e10T^{2} \) |
| 19 | \( 1 + 8.37e3T + 3.22e11T^{2} \) |
| 23 | \( 1 + 3.64e5T + 1.80e12T^{2} \) |
| 29 | \( 1 - 3.50e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 5.20e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 4.99e5T + 1.29e14T^{2} \) |
| 41 | \( 1 - 5.43e6T + 3.27e14T^{2} \) |
| 43 | \( 1 + 3.54e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 1.21e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 1.04e8T + 3.29e15T^{2} \) |
| 59 | \( 1 + 4.16e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 5.67e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 1.74e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 3.46e7T + 4.58e16T^{2} \) |
| 73 | \( 1 + 3.93e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 1.85e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 3.62e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 5.04e7T + 3.50e17T^{2} \) |
| 97 | \( 1 + 9.67e8T + 7.60e17T^{2} \) |
| 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
−11.31166027439041554500591492841, −9.779745959018739915285819353444, −8.585232882630260016723525882766, −7.43782701448976235074314330550, −6.23501109542755251686272402698, −5.01636395233291474512521021038, −4.26708327189998892049671130130, −3.20275844153308335655405315631, −1.71864682062221643537832366529, 0,
1.71864682062221643537832366529, 3.20275844153308335655405315631, 4.26708327189998892049671130130, 5.01636395233291474512521021038, 6.23501109542755251686272402698, 7.43782701448976235074314330550, 8.585232882630260016723525882766, 9.779745959018739915285819353444, 11.31166027439041554500591492841
