| L(s) = 1 | + 34.3·2-s − 129.·3-s + 665.·4-s − 1.07e3·5-s − 4.43e3·6-s + 9.33e3·7-s + 5.27e3·8-s − 2.95e3·9-s − 3.69e4·10-s − 5.80e4·11-s − 8.61e4·12-s − 1.08e5·13-s + 3.20e5·14-s + 1.39e5·15-s − 1.59e5·16-s − 6.68e5·17-s − 1.01e5·18-s − 2.43e5·19-s − 7.16e5·20-s − 1.20e6·21-s − 1.99e6·22-s + 2.32e6·23-s − 6.82e5·24-s − 7.94e5·25-s − 3.70e6·26-s + 2.92e6·27-s + 6.21e6·28-s + ⋯ |
| L(s) = 1 | + 1.51·2-s − 0.921·3-s + 1.30·4-s − 0.770·5-s − 1.39·6-s + 1.46·7-s + 0.455·8-s − 0.150·9-s − 1.16·10-s − 1.19·11-s − 1.19·12-s − 1.04·13-s + 2.22·14-s + 0.709·15-s − 0.609·16-s − 1.94·17-s − 0.228·18-s − 0.428·19-s − 1.00·20-s − 1.35·21-s − 1.81·22-s + 1.73·23-s − 0.420·24-s − 0.406·25-s − 1.59·26-s + 1.06·27-s + 1.91·28-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 | \( 1 - 34.3T + 512T^{2} \) |
| 3 | \( 1 + 129.T + 1.96e4T^{2} \) |
| 5 | \( 1 + 1.07e3T + 1.95e6T^{2} \) |
| 7 | \( 1 - 9.33e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 5.80e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 1.08e5T + 1.06e10T^{2} \) |
| 17 | \( 1 + 6.68e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 2.43e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 2.32e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 2.01e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 8.33e6T + 2.64e13T^{2} \) |
| 41 | \( 1 + 7.34e6T + 3.27e14T^{2} \) |
| 43 | \( 1 + 8.69e6T + 5.02e14T^{2} \) |
| 47 | \( 1 + 4.36e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 2.04e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 1.49e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 1.03e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 1.75e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 6.24e7T + 4.58e16T^{2} \) |
| 73 | \( 1 - 4.17e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 2.96e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 8.94e7T + 1.86e17T^{2} \) |
| 89 | \( 1 + 9.24e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 6.81e8T + 7.60e17T^{2} \) |
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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
−13.71234068564818883039712512690, −12.49830550085026192585213168678, −11.46426953394530945577854677135, −10.95590125978649549453863946721, −8.270659765020233688091925880443, −6.72052522663558475410107086142, −5.00088335354391763628875340276, −4.70010643160135096236676517739, −2.55102539853698195198170479651, 0,
2.55102539853698195198170479651, 4.70010643160135096236676517739, 5.00088335354391763628875340276, 6.72052522663558475410107086142, 8.270659765020233688091925880443, 10.95590125978649549453863946721, 11.46426953394530945577854677135, 12.49830550085026192585213168678, 13.71234068564818883039712512690
