| L(s) = 1 | − 3.19·2-s + 133.·3-s − 501.·4-s + 16.5·5-s − 426.·6-s + 4.31e3·7-s + 3.23e3·8-s − 1.79e3·9-s − 52.9·10-s − 6.12e4·11-s − 6.71e4·12-s + 2.97e4·13-s − 1.37e4·14-s + 2.21e3·15-s + 2.46e5·16-s − 4.72e5·17-s + 5.73e3·18-s + 1.87e5·19-s − 8.32e3·20-s + 5.77e5·21-s + 1.95e5·22-s − 1.42e6·23-s + 4.32e5·24-s − 1.95e6·25-s − 9.49e4·26-s − 2.87e6·27-s − 2.16e6·28-s + ⋯ |
| L(s) = 1 | − 0.141·2-s + 0.953·3-s − 0.980·4-s + 0.0118·5-s − 0.134·6-s + 0.679·7-s + 0.279·8-s − 0.0913·9-s − 0.00167·10-s − 1.26·11-s − 0.934·12-s + 0.288·13-s − 0.0958·14-s + 0.0113·15-s + 0.940·16-s − 1.37·17-s + 0.0128·18-s + 0.329·19-s − 0.0116·20-s + 0.647·21-s + 0.177·22-s − 1.06·23-s + 0.266·24-s − 0.999·25-s − 0.0407·26-s − 1.04·27-s − 0.666·28-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 | \( 1 + 3.19T + 512T^{2} \) |
| 3 | \( 1 - 133.T + 1.96e4T^{2} \) |
| 5 | \( 1 - 16.5T + 1.95e6T^{2} \) |
| 7 | \( 1 - 4.31e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 6.12e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 2.97e4T + 1.06e10T^{2} \) |
| 17 | \( 1 + 4.72e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 1.87e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.42e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 1.65e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 4.87e6T + 2.64e13T^{2} \) |
| 41 | \( 1 - 1.04e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 2.08e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 4.39e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 4.21e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 1.24e8T + 8.66e15T^{2} \) |
| 61 | \( 1 - 8.86e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 2.64e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 5.43e7T + 4.58e16T^{2} \) |
| 73 | \( 1 + 2.29e7T + 5.88e16T^{2} \) |
| 79 | \( 1 + 1.60e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 6.09e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 9.44e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 4.94e8T + 7.60e17T^{2} \) |
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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
−13.78534957955237294629584930663, −13.01161879261053412762510862890, −11.16893919628271858423443363216, −9.712279691102537095758954524097, −8.542696738555844808850663666677, −7.80407085133656508521180406289, −5.40856226876968065136866271795, −3.91822948828322876533145225694, −2.16953662266911632620252124714, 0,
2.16953662266911632620252124714, 3.91822948828322876533145225694, 5.40856226876968065136866271795, 7.80407085133656508521180406289, 8.542696738555844808850663666677, 9.712279691102537095758954524097, 11.16893919628271858423443363216, 13.01161879261053412762510862890, 13.78534957955237294629584930663
