| L(s) = 1 | + 115.·2-s + 6.20e3·3-s − 1.17e5·4-s − 3.90e5·5-s + 7.14e5·6-s − 6.90e5·7-s − 2.86e7·8-s − 9.06e7·9-s − 4.49e7·10-s − 3.25e8·11-s − 7.30e8·12-s − 5.16e9·13-s − 7.94e7·14-s − 2.42e9·15-s + 1.21e10·16-s + 1.15e10·17-s − 1.04e10·18-s + 8.37e10·19-s + 4.60e10·20-s − 4.27e9·21-s − 3.74e10·22-s + 1.73e10·23-s − 1.77e11·24-s + 1.52e11·25-s − 5.94e11·26-s − 1.36e12·27-s + 8.12e10·28-s + ⋯ |
| L(s) = 1 | + 0.318·2-s + 0.545·3-s − 0.898·4-s − 0.447·5-s + 0.173·6-s − 0.0452·7-s − 0.604·8-s − 0.702·9-s − 0.142·10-s − 0.457·11-s − 0.490·12-s − 1.75·13-s − 0.0143·14-s − 0.244·15-s + 0.706·16-s + 0.400·17-s − 0.223·18-s + 1.13·19-s + 0.401·20-s − 0.0246·21-s − 0.145·22-s + 0.0463·23-s − 0.329·24-s + 0.200·25-s − 0.558·26-s − 0.928·27-s + 0.0406·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(9)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{19}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 3.90e5T \) |
| good | 2 | \( 1 - 115.T + 1.31e5T^{2} \) |
| 3 | \( 1 - 6.20e3T + 1.29e8T^{2} \) |
| 7 | \( 1 + 6.90e5T + 2.32e14T^{2} \) |
| 11 | \( 1 + 3.25e8T + 5.05e17T^{2} \) |
| 13 | \( 1 + 5.16e9T + 8.65e18T^{2} \) |
| 17 | \( 1 - 1.15e10T + 8.27e20T^{2} \) |
| 19 | \( 1 - 8.37e10T + 5.48e21T^{2} \) |
| 23 | \( 1 - 1.73e10T + 1.41e23T^{2} \) |
| 29 | \( 1 + 1.75e12T + 7.25e24T^{2} \) |
| 31 | \( 1 - 5.43e12T + 2.25e25T^{2} \) |
| 37 | \( 1 - 2.65e12T + 4.56e26T^{2} \) |
| 41 | \( 1 + 7.37e13T + 2.61e27T^{2} \) |
| 43 | \( 1 + 7.57e13T + 5.87e27T^{2} \) |
| 47 | \( 1 - 2.09e14T + 2.66e28T^{2} \) |
| 53 | \( 1 + 4.10e14T + 2.05e29T^{2} \) |
| 59 | \( 1 + 2.25e15T + 1.27e30T^{2} \) |
| 61 | \( 1 + 9.64e14T + 2.24e30T^{2} \) |
| 67 | \( 1 - 4.11e15T + 1.10e31T^{2} \) |
| 71 | \( 1 - 6.31e15T + 2.96e31T^{2} \) |
| 73 | \( 1 + 5.15e15T + 4.74e31T^{2} \) |
| 79 | \( 1 + 9.84e15T + 1.81e32T^{2} \) |
| 83 | \( 1 - 2.23e16T + 4.21e32T^{2} \) |
| 89 | \( 1 - 5.52e16T + 1.37e33T^{2} \) |
| 97 | \( 1 + 1.47e17T + 5.95e33T^{2} \) |
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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
−18.86018570440623316264336095935, −17.18097010012877037048036990315, −15.01813675310473101396031301312, −13.83384473435392049460259104332, −12.12845439585492633111187849924, −9.601654067456193616680699061606, −7.923514749749207531251585691270, −5.06116963598790235535915510382, −3.08856869044528160566801365891, 0,
3.08856869044528160566801365891, 5.06116963598790235535915510382, 7.923514749749207531251585691270, 9.601654067456193616680699061606, 12.12845439585492633111187849924, 13.83384473435392049460259104332, 15.01813675310473101396031301312, 17.18097010012877037048036990315, 18.86018570440623316264336095935
