| L(s) = 1 | − 3.93·2-s − 112.·4-s + 305.·5-s − 1.40e3·7-s + 946.·8-s − 1.20e3·10-s − 322.·11-s − 6.38e3·13-s + 5.53e3·14-s + 1.06e4·16-s + 1.48e4·17-s − 1.27e4·19-s − 3.43e4·20-s + 1.26e3·22-s + 1.73e4·23-s + 1.52e4·25-s + 2.51e4·26-s + 1.58e5·28-s + 6.86e4·29-s + 1.53e5·31-s − 1.63e5·32-s − 5.84e4·34-s − 4.30e5·35-s + 1.98e5·37-s + 4.99e4·38-s + 2.89e5·40-s − 4.19e5·41-s + ⋯ |
| L(s) = 1 | − 0.347·2-s − 0.879·4-s + 1.09·5-s − 1.55·7-s + 0.653·8-s − 0.380·10-s − 0.0729·11-s − 0.806·13-s + 0.538·14-s + 0.652·16-s + 0.733·17-s − 0.425·19-s − 0.961·20-s + 0.0253·22-s + 0.296·23-s + 0.195·25-s + 0.280·26-s + 1.36·28-s + 0.523·29-s + 0.923·31-s − 0.879·32-s − 0.254·34-s − 1.69·35-s + 0.642·37-s + 0.147·38-s + 0.714·40-s − 0.949·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 59 | \( 1 + 2.05e5T \) |
| good | 2 | \( 1 + 3.93T + 128T^{2} \) |
| 5 | \( 1 - 305.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 1.40e3T + 8.23e5T^{2} \) |
| 11 | \( 1 + 322.T + 1.94e7T^{2} \) |
| 13 | \( 1 + 6.38e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 1.48e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 1.27e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 1.73e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 6.86e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.53e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 1.98e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 4.19e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 1.04e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 7.38e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 2.48e5T + 1.17e12T^{2} \) |
| 61 | \( 1 - 9.15e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + 8.28e5T + 6.06e12T^{2} \) |
| 71 | \( 1 - 3.02e5T + 9.09e12T^{2} \) |
| 73 | \( 1 + 3.36e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 6.08e5T + 1.92e13T^{2} \) |
| 83 | \( 1 - 3.11e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 8.70e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.80e6T + 8.07e13T^{2} \) |
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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
−9.473748646815403378670615584968, −8.649660866870739617610655988099, −7.48745377361531544251592087855, −6.43042207329571029513491249690, −5.66226504619156703647955650242, −4.64245425889115840634969727039, −3.41100530100252675330617967640, −2.40179886841308789227811996522, −1.00507557365110119264674575032, 0,
1.00507557365110119264674575032, 2.40179886841308789227811996522, 3.41100530100252675330617967640, 4.64245425889115840634969727039, 5.66226504619156703647955650242, 6.43042207329571029513491249690, 7.48745377361531544251592087855, 8.649660866870739617610655988099, 9.473748646815403378670615584968
