| L(s) = 1 | − 4·2-s + 19.0·3-s + 16·4-s + 7.20·5-s − 76.2·6-s + 53.6·7-s − 64·8-s + 120.·9-s − 28.8·10-s − 239.·11-s + 305.·12-s − 214.·14-s + 137.·15-s + 256·16-s − 1.97e3·17-s − 482.·18-s + 373.·19-s + 115.·20-s + 1.02e3·21-s + 958.·22-s + 51.4·23-s − 1.22e3·24-s − 3.07e3·25-s − 2.33e3·27-s + 857.·28-s + 4.79e3·29-s − 549.·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.22·3-s + 0.5·4-s + 0.128·5-s − 0.864·6-s + 0.413·7-s − 0.353·8-s + 0.496·9-s − 0.0911·10-s − 0.597·11-s + 0.611·12-s − 0.292·14-s + 0.157·15-s + 0.250·16-s − 1.65·17-s − 0.350·18-s + 0.237·19-s + 0.0644·20-s + 0.505·21-s + 0.422·22-s + 0.0202·23-s − 0.432·24-s − 0.983·25-s − 0.616·27-s + 0.206·28-s + 1.05·29-s − 0.111·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 4T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - 19.0T + 243T^{2} \) |
| 5 | \( 1 - 7.20T + 3.12e3T^{2} \) |
| 7 | \( 1 - 53.6T + 1.68e4T^{2} \) |
| 11 | \( 1 + 239.T + 1.61e5T^{2} \) |
| 17 | \( 1 + 1.97e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 373.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 51.4T + 6.43e6T^{2} \) |
| 29 | \( 1 - 4.79e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 6.90e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.14e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.25e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.15e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.86e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 9.31e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 5.06e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 5.42e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.02e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 6.52e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 6.85e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 1.06e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 2.35e3T + 3.93e9T^{2} \) |
| 89 | \( 1 - 9.36e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 3.11e4T + 8.58e9T^{2} \) |
| 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
−10.07991657122364246337358562580, −9.053923761128152583222504298828, −8.525852876194197072812115536796, −7.68889830868501058980082796049, −6.73258636950530175302554814160, −5.29423589014569143371562655113, −3.84629455008342125298583691251, −2.58677046030454400668603076499, −1.79127916766634891950069890201, 0,
1.79127916766634891950069890201, 2.58677046030454400668603076499, 3.84629455008342125298583691251, 5.29423589014569143371562655113, 6.73258636950530175302554814160, 7.68889830868501058980082796049, 8.525852876194197072812115536796, 9.053923761128152583222504298828, 10.07991657122364246337358562580
