| L(s) = 1 | − 8·2-s − 19.6·3-s + 64·4-s − 242.·5-s + 157.·6-s − 1.44e3·7-s − 512·8-s − 1.80e3·9-s + 1.93e3·10-s − 2.21e3·11-s − 1.25e3·12-s − 1.40e4·13-s + 1.15e4·14-s + 4.76e3·15-s + 4.09e3·16-s − 1.99e4·17-s + 1.44e4·18-s + 3.07e4·19-s − 1.55e4·20-s + 2.83e4·21-s + 1.77e4·22-s + 6.63e4·23-s + 1.00e4·24-s − 1.94e4·25-s + 1.12e5·26-s + 7.84e4·27-s − 9.22e4·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.420·3-s + 0.5·4-s − 0.866·5-s + 0.297·6-s − 1.58·7-s − 0.353·8-s − 0.823·9-s + 0.612·10-s − 0.502·11-s − 0.210·12-s − 1.77·13-s + 1.12·14-s + 0.364·15-s + 0.250·16-s − 0.986·17-s + 0.582·18-s + 1.02·19-s − 0.433·20-s + 0.667·21-s + 0.355·22-s + 1.13·23-s + 0.148·24-s − 0.248·25-s + 1.25·26-s + 0.766·27-s − 0.794·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 8T \) |
| 269 | \( 1 - 1.94e7T \) |
| good | 3 | \( 1 + 19.6T + 2.18e3T^{2} \) |
| 5 | \( 1 + 242.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 1.44e3T + 8.23e5T^{2} \) |
| 11 | \( 1 + 2.21e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 1.40e4T + 6.27e7T^{2} \) |
| 17 | \( 1 + 1.99e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 3.07e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 6.63e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 2.43e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 5.66e4T + 2.75e10T^{2} \) |
| 37 | \( 1 + 5.31e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 3.59e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 5.52e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 2.99e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 2.10e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 2.94e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 2.89e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 1.08e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 4.83e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 4.65e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 6.00e5T + 1.92e13T^{2} \) |
| 83 | \( 1 - 4.21e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 6.58e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.36e7T + 8.07e13T^{2} \) |
| 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
−9.297188512502135514457579846562, −8.442917379104110740816191548732, −7.30096373562027807376639611486, −6.82760036111758859431930650546, −5.69562166299809962121910497104, −4.62797919893298041782563168210, −3.11195367033617062222828259717, −2.62889722120193749705310329577, −0.61389306659533056323248079613, 0,
0.61389306659533056323248079613, 2.62889722120193749705310329577, 3.11195367033617062222828259717, 4.62797919893298041782563168210, 5.69562166299809962121910497104, 6.82760036111758859431930650546, 7.30096373562027807376639611486, 8.442917379104110740816191548732, 9.297188512502135514457579846562
