| L(s) = 1 | + 2·2-s + 3·3-s + 4·4-s − 12.8·5-s + 6·6-s − 7·7-s + 8·8-s + 9·9-s − 25.7·10-s + 11.2·11-s + 12·12-s − 2.52·13-s − 14·14-s − 38.6·15-s + 16·16-s + 32.4·17-s + 18·18-s − 44.8·19-s − 51.5·20-s − 21·21-s + 22.4·22-s + 23·23-s + 24·24-s + 41.2·25-s − 5.04·26-s + 27·27-s − 28·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.15·5-s + 0.408·6-s − 0.377·7-s + 0.353·8-s + 0.333·9-s − 0.815·10-s + 0.307·11-s + 0.288·12-s − 0.0537·13-s − 0.267·14-s − 0.665·15-s + 0.250·16-s + 0.463·17-s + 0.235·18-s − 0.541·19-s − 0.576·20-s − 0.218·21-s + 0.217·22-s + 0.208·23-s + 0.204·24-s + 0.330·25-s − 0.0380·26-s + 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 3 | \( 1 - 3T \) |
| 7 | \( 1 + 7T \) |
| 23 | \( 1 - 23T \) |
| good | 5 | \( 1 + 12.8T + 125T^{2} \) |
| 11 | \( 1 - 11.2T + 1.33e3T^{2} \) |
| 13 | \( 1 + 2.52T + 2.19e3T^{2} \) |
| 17 | \( 1 - 32.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 44.8T + 6.85e3T^{2} \) |
| 29 | \( 1 + 241.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 13.9T + 2.97e4T^{2} \) |
| 37 | \( 1 + 93.1T + 5.06e4T^{2} \) |
| 41 | \( 1 + 388.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 272.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 394.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 491.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 759.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 709.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 708.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 160.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 195.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.22e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 912.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 939.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 304.T + 9.12e5T^{2} \) |
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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
−9.124677746886052458910265443189, −8.250002806386322561502581764224, −7.47263870460967963028731502503, −6.78881959339226872604065212610, −5.67460945519474960980744544081, −4.53325587212509493110211523622, −3.73137025613922975732278568262, −3.08952726392794489718810848493, −1.70143948803794783762915725341, 0,
1.70143948803794783762915725341, 3.08952726392794489718810848493, 3.73137025613922975732278568262, 4.53325587212509493110211523622, 5.67460945519474960980744544081, 6.78881959339226872604065212610, 7.47263870460967963028731502503, 8.250002806386322561502581764224, 9.124677746886052458910265443189
