| L(s) = 1 | + 2.60·2-s + 20.6·3-s − 25.2·4-s + 25·5-s + 53.7·6-s − 10.8·7-s − 149.·8-s + 183.·9-s + 65.1·10-s + 121·11-s − 520.·12-s + 263.·13-s − 28.3·14-s + 516.·15-s + 418.·16-s − 247.·17-s + 477.·18-s − 361·19-s − 630.·20-s − 224.·21-s + 315.·22-s − 1.49e3·23-s − 3.07e3·24-s + 625·25-s + 687.·26-s − 1.23e3·27-s + 274.·28-s + ⋯ |
| L(s) = 1 | + 0.460·2-s + 1.32·3-s − 0.787·4-s + 0.447·5-s + 0.609·6-s − 0.0840·7-s − 0.823·8-s + 0.753·9-s + 0.205·10-s + 0.301·11-s − 1.04·12-s + 0.432·13-s − 0.0386·14-s + 0.592·15-s + 0.408·16-s − 0.207·17-s + 0.347·18-s − 0.229·19-s − 0.352·20-s − 0.111·21-s + 0.138·22-s − 0.589·23-s − 1.09·24-s + 0.200·25-s + 0.199·26-s − 0.326·27-s + 0.0661·28-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 | \( 1 - 2.60T + 32T^{2} \) |
| 3 | \( 1 - 20.6T + 243T^{2} \) |
| 7 | \( 1 + 10.8T + 1.68e4T^{2} \) |
| 13 | \( 1 - 263.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 247.T + 1.41e6T^{2} \) |
| 23 | \( 1 + 1.49e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 1.84e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 3.39e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 7.85e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.48e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 9.30e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.73e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.77e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 4.17e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.45e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 4.04e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 4.06e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 7.03e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 4.50e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 3.14e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 4.06e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 3.20e4T + 8.58e9T^{2} \) |
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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
−8.712368265204017138286504469983, −8.297365563404447092817395834536, −7.18252359441849273079909500542, −6.09961867657461491682302221128, −5.24547753016041646016707976484, −4.09561239287030050487729051626, −3.52601157515194328388669100833, −2.54524447834400654939604186392, −1.49745710415165846936323718062, 0,
1.49745710415165846936323718062, 2.54524447834400654939604186392, 3.52601157515194328388669100833, 4.09561239287030050487729051626, 5.24547753016041646016707976484, 6.09961867657461491682302221128, 7.18252359441849273079909500542, 8.297365563404447092817395834536, 8.712368265204017138286504469983
