| L(s) = 1 | + 24.7·2-s − 84.2·3-s + 101.·4-s + 1.37e3·5-s − 2.08e3·6-s − 3.15e3·7-s − 1.01e4·8-s − 1.25e4·9-s + 3.40e4·10-s + 1.41e4·11-s − 8.55e3·12-s − 1.16e4·13-s − 7.81e4·14-s − 1.15e5·15-s − 3.03e5·16-s − 1.87e5·17-s − 3.11e5·18-s − 7.01e5·19-s + 1.39e5·20-s + 2.65e5·21-s + 3.50e5·22-s − 2.48e6·23-s + 8.56e5·24-s − 6.70e4·25-s − 2.87e5·26-s + 2.71e6·27-s − 3.20e5·28-s + ⋯ |
| L(s) = 1 | + 1.09·2-s − 0.600·3-s + 0.198·4-s + 0.982·5-s − 0.657·6-s − 0.496·7-s − 0.877·8-s − 0.639·9-s + 1.07·10-s + 0.291·11-s − 0.119·12-s − 0.112·13-s − 0.543·14-s − 0.589·15-s − 1.15·16-s − 0.544·17-s − 0.700·18-s − 1.23·19-s + 0.194·20-s + 0.298·21-s + 0.318·22-s − 1.85·23-s + 0.526·24-s − 0.0343·25-s − 0.123·26-s + 0.984·27-s − 0.0985·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 + 1.87e6T \) |
| good | 2 | \( 1 - 24.7T + 512T^{2} \) |
| 3 | \( 1 + 84.2T + 1.96e4T^{2} \) |
| 5 | \( 1 - 1.37e3T + 1.95e6T^{2} \) |
| 7 | \( 1 + 3.15e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 1.41e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 1.16e4T + 1.06e10T^{2} \) |
| 17 | \( 1 + 1.87e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 7.01e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 2.48e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 9.35e5T + 1.45e13T^{2} \) |
| 31 | \( 1 - 2.47e6T + 2.64e13T^{2} \) |
| 41 | \( 1 - 1.55e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 1.48e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 2.67e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 4.04e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 4.81e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 1.42e8T + 1.16e16T^{2} \) |
| 67 | \( 1 + 2.59e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 1.71e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 1.29e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 5.94e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 8.91e7T + 1.86e17T^{2} \) |
| 89 | \( 1 + 2.13e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 1.71e8T + 7.60e17T^{2} \) |
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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
−13.74812442658812934482148666677, −12.72917332851334549804037408715, −11.67907012287381864499481138852, −10.15878288334520538114727883618, −8.809427303911649878657378564034, −6.34015167875293831380520706180, −5.75024630842334317229789142167, −4.22732028993501521076732691672, −2.44238278700291967148090768601, 0,
2.44238278700291967148090768601, 4.22732028993501521076732691672, 5.75024630842334317229789142167, 6.34015167875293831380520706180, 8.809427303911649878657378564034, 10.15878288334520538114727883618, 11.67907012287381864499481138852, 12.72917332851334549804037408715, 13.74812442658812934482148666677
