| L(s) = 1 | + 1.32·2-s − 9.29·3-s − 6.25·4-s + 7.30·5-s − 12.2·6-s − 30.2·7-s − 18.8·8-s + 59.4·9-s + 9.64·10-s + 37.2·11-s + 58.1·12-s − 37.0·13-s − 40.0·14-s − 67.8·15-s + 25.1·16-s − 0.214·17-s + 78.5·18-s − 108.·19-s − 45.6·20-s + 281.·21-s + 49.2·22-s − 123.·23-s + 175.·24-s − 71.6·25-s − 48.8·26-s − 301.·27-s + 189.·28-s + ⋯ |
| L(s) = 1 | + 0.467·2-s − 1.78·3-s − 0.781·4-s + 0.653·5-s − 0.835·6-s − 1.63·7-s − 0.832·8-s + 2.20·9-s + 0.305·10-s + 1.02·11-s + 1.39·12-s − 0.789·13-s − 0.764·14-s − 1.16·15-s + 0.393·16-s − 0.00305·17-s + 1.02·18-s − 1.30·19-s − 0.510·20-s + 2.92·21-s + 0.477·22-s − 1.11·23-s + 1.48·24-s − 0.573·25-s − 0.368·26-s − 2.14·27-s + 1.27·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 37 | \( 1 - 37T \) |
| good | 2 | \( 1 - 1.32T + 8T^{2} \) |
| 3 | \( 1 + 9.29T + 27T^{2} \) |
| 5 | \( 1 - 7.30T + 125T^{2} \) |
| 7 | \( 1 + 30.2T + 343T^{2} \) |
| 11 | \( 1 - 37.2T + 1.33e3T^{2} \) |
| 13 | \( 1 + 37.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + 0.214T + 4.91e3T^{2} \) |
| 19 | \( 1 + 108.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 123.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 15.0T + 2.43e4T^{2} \) |
| 31 | \( 1 - 157.T + 2.97e4T^{2} \) |
| 41 | \( 1 + 196.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 40.9T + 7.95e4T^{2} \) |
| 47 | \( 1 - 389.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 19.6T + 1.48e5T^{2} \) |
| 59 | \( 1 + 681.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 731.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 527.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 226.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 43.3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 230.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.39e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 272.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.22e3T + 9.12e5T^{2} \) |
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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
−15.48859339069440497513198449219, −13.79322324480982996309182627966, −12.66270797803784377492667939926, −12.05840553486732605241689984041, −10.23873620884608393872252679169, −9.462703982478116740296473652350, −6.52302048145306040180669358089, −5.86282248255721465636512227275, −4.24568600511090246463778800004, 0,
4.24568600511090246463778800004, 5.86282248255721465636512227275, 6.52302048145306040180669358089, 9.462703982478116740296473652350, 10.23873620884608393872252679169, 12.05840553486732605241689984041, 12.66270797803784377492667939926, 13.79322324480982996309182627966, 15.48859339069440497513198449219
