| L(s) = 1 | + 8.15·2-s + 19.4·3-s + 34.5·4-s + 25·5-s + 158.·6-s − 80.0·7-s + 20.7·8-s + 136.·9-s + 203.·10-s − 121·11-s + 672.·12-s − 517.·13-s − 653.·14-s + 486.·15-s − 936.·16-s + 119.·17-s + 1.11e3·18-s + 361·19-s + 863.·20-s − 1.55e3·21-s − 987.·22-s − 2.11e3·23-s + 404.·24-s + 625·25-s − 4.22e3·26-s − 2.08e3·27-s − 2.76e3·28-s + ⋯ |
| L(s) = 1 | + 1.44·2-s + 1.24·3-s + 1.07·4-s + 0.447·5-s + 1.80·6-s − 0.617·7-s + 0.114·8-s + 0.560·9-s + 0.644·10-s − 0.301·11-s + 1.34·12-s − 0.849·13-s − 0.890·14-s + 0.558·15-s − 0.914·16-s + 0.100·17-s + 0.808·18-s + 0.229·19-s + 0.482·20-s − 0.771·21-s − 0.434·22-s − 0.833·23-s + 0.143·24-s + 0.200·25-s − 1.22·26-s − 0.549·27-s − 0.666·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - 25T \) |
| 11 | \( 1 + 121T \) |
| 19 | \( 1 - 361T \) |
| good | 2 | \( 1 - 8.15T + 32T^{2} \) |
| 3 | \( 1 - 19.4T + 243T^{2} \) |
| 7 | \( 1 + 80.0T + 1.68e4T^{2} \) |
| 13 | \( 1 + 517.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 119.T + 1.41e6T^{2} \) |
| 23 | \( 1 + 2.11e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 1.97e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 6.75e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.46e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 3.53e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 4.92e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.19e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.52e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 4.96e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.36e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.67e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 2.84e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 6.65e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 8.67e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 8.62e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 7.68e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 3.50e4T + 8.58e9T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.909024781957883699897201584067, −7.78400571910492285770324512534, −7.01639486850529966882986009612, −5.96799028040143529838980355561, −5.28589269886317725156786864560, −4.18540997812837159541320594683, −3.40315673029094942691456393019, −2.66660930410760273553098139451, −1.95238121897286064016425805301, 0,
1.95238121897286064016425805301, 2.66660930410760273553098139451, 3.40315673029094942691456393019, 4.18540997812837159541320594683, 5.28589269886317725156786864560, 5.96799028040143529838980355561, 7.01639486850529966882986009612, 7.78400571910492285770324512534, 8.909024781957883699897201584067
