| L(s) = 1 | − 12.3·2-s − 116.·3-s − 359.·4-s + 44.3·5-s + 1.43e3·6-s + 8.40e3·7-s + 1.07e4·8-s − 6.20e3·9-s − 546.·10-s + 7.72e4·11-s + 4.17e4·12-s − 1.63e5·13-s − 1.03e5·14-s − 5.14e3·15-s + 5.13e4·16-s + 4.23e4·17-s + 7.65e4·18-s + 1.75e5·19-s − 1.59e4·20-s − 9.75e5·21-s − 9.53e5·22-s − 6.01e5·23-s − 1.24e6·24-s − 1.95e6·25-s + 2.01e6·26-s + 3.00e6·27-s − 3.02e6·28-s + ⋯ |
| L(s) = 1 | − 0.545·2-s − 0.827·3-s − 0.702·4-s + 0.0317·5-s + 0.451·6-s + 1.32·7-s + 0.928·8-s − 0.314·9-s − 0.0172·10-s + 1.59·11-s + 0.581·12-s − 1.58·13-s − 0.721·14-s − 0.0262·15-s + 0.196·16-s + 0.123·17-s + 0.171·18-s + 0.308·19-s − 0.0222·20-s − 1.09·21-s − 0.867·22-s − 0.448·23-s − 0.768·24-s − 0.998·25-s + 0.865·26-s + 1.08·27-s − 0.929·28-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 | \( 1 + 12.3T + 512T^{2} \) |
| 3 | \( 1 + 116.T + 1.96e4T^{2} \) |
| 5 | \( 1 - 44.3T + 1.95e6T^{2} \) |
| 7 | \( 1 - 8.40e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 7.72e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 1.63e5T + 1.06e10T^{2} \) |
| 17 | \( 1 - 4.23e4T + 1.18e11T^{2} \) |
| 19 | \( 1 - 1.75e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 6.01e5T + 1.80e12T^{2} \) |
| 29 | \( 1 + 1.03e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 3.92e6T + 2.64e13T^{2} \) |
| 41 | \( 1 + 2.36e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 1.00e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 3.13e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 3.78e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 1.37e8T + 8.66e15T^{2} \) |
| 61 | \( 1 + 9.98e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 4.50e7T + 2.72e16T^{2} \) |
| 71 | \( 1 + 2.30e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 3.71e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 1.86e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 7.36e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 2.30e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 1.37e9T + 7.60e17T^{2} \) |
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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
−14.05337792235618353986241685780, −12.15926998198001751105560928668, −11.38038955237894822860010814705, −9.949882183238079369537684220878, −8.734542852720572499765607655268, −7.39066014485179406001562491913, −5.49232738074010621646785416268, −4.35165554778423436660424512217, −1.51829705786988136337311170308, 0,
1.51829705786988136337311170308, 4.35165554778423436660424512217, 5.49232738074010621646785416268, 7.39066014485179406001562491913, 8.734542852720572499765607655268, 9.949882183238079369537684220878, 11.38038955237894822860010814705, 12.15926998198001751105560928668, 14.05337792235618353986241685780
