| L(s) = 1 | − 145.·2-s + 254.·3-s − 1.16e4·4-s − 2.33e5·5-s − 3.70e4·6-s + 2.20e6·7-s + 6.45e6·8-s − 1.42e7·9-s + 3.39e7·10-s + 9.38e7·11-s − 2.95e6·12-s − 1.18e8·13-s − 3.20e8·14-s − 5.94e7·15-s − 5.58e8·16-s − 1.65e8·17-s + 2.07e9·18-s + 7.03e9·19-s + 2.70e9·20-s + 5.62e8·21-s − 1.36e10·22-s + 3.40e9·23-s + 1.64e9·24-s + 2.38e10·25-s + 1.73e10·26-s − 7.30e9·27-s − 2.55e10·28-s + ⋯ |
| L(s) = 1 | − 0.803·2-s + 0.0673·3-s − 0.354·4-s − 1.33·5-s − 0.0540·6-s + 1.01·7-s + 1.08·8-s − 0.995·9-s + 1.07·10-s + 1.45·11-s − 0.0238·12-s − 0.525·13-s − 0.813·14-s − 0.0898·15-s − 0.520·16-s − 0.0978·17-s + 0.799·18-s + 1.80·19-s + 0.472·20-s + 0.0681·21-s − 1.16·22-s + 0.208·23-s + 0.0732·24-s + 0.780·25-s + 0.422·26-s − 0.134·27-s − 0.358·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(8)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{17}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 - 3.40e9T \) |
| good | 2 | \( 1 + 145.T + 3.27e4T^{2} \) |
| 3 | \( 1 - 254.T + 1.43e7T^{2} \) |
| 5 | \( 1 + 2.33e5T + 3.05e10T^{2} \) |
| 7 | \( 1 - 2.20e6T + 4.74e12T^{2} \) |
| 11 | \( 1 - 9.38e7T + 4.17e15T^{2} \) |
| 13 | \( 1 + 1.18e8T + 5.11e16T^{2} \) |
| 17 | \( 1 + 1.65e8T + 2.86e18T^{2} \) |
| 19 | \( 1 - 7.03e9T + 1.51e19T^{2} \) |
| 29 | \( 1 + 2.60e10T + 8.62e21T^{2} \) |
| 31 | \( 1 + 1.38e11T + 2.34e22T^{2} \) |
| 37 | \( 1 - 1.31e11T + 3.33e23T^{2} \) |
| 41 | \( 1 + 4.64e11T + 1.55e24T^{2} \) |
| 43 | \( 1 + 2.76e12T + 3.17e24T^{2} \) |
| 47 | \( 1 - 8.88e11T + 1.20e25T^{2} \) |
| 53 | \( 1 + 3.03e12T + 7.31e25T^{2} \) |
| 59 | \( 1 - 3.46e11T + 3.65e26T^{2} \) |
| 61 | \( 1 + 1.69e13T + 6.02e26T^{2} \) |
| 67 | \( 1 + 3.26e13T + 2.46e27T^{2} \) |
| 71 | \( 1 - 4.12e13T + 5.87e27T^{2} \) |
| 73 | \( 1 - 1.12e14T + 8.90e27T^{2} \) |
| 79 | \( 1 + 2.27e14T + 2.91e28T^{2} \) |
| 83 | \( 1 - 8.49e13T + 6.11e28T^{2} \) |
| 89 | \( 1 - 2.27e13T + 1.74e29T^{2} \) |
| 97 | \( 1 + 1.34e15T + 6.33e29T^{2} \) |
| 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
−13.99708640540049445069170655976, −11.88964895740884051163482790528, −11.23405264263779118719820285814, −9.363350987796462961849457181136, −8.308966134188826285952977319063, −7.37574866444770424211062445685, −4.97553977308964480331055665045, −3.62916195048457276141581664652, −1.30481063773375726953460593502, 0,
1.30481063773375726953460593502, 3.62916195048457276141581664652, 4.97553977308964480331055665045, 7.37574866444770424211062445685, 8.308966134188826285952977319063, 9.363350987796462961849457181136, 11.23405264263779118719820285814, 11.88964895740884051163482790528, 13.99708640540049445069170655976
