| L(s) = 1 | + 8.18·2-s − 61.0·4-s − 131.·5-s − 311.·7-s − 1.54e3·8-s − 1.07e3·10-s − 3.33e3·11-s + 9.94e3·13-s − 2.54e3·14-s − 4.84e3·16-s + 2.14e4·17-s + 6.39e3·19-s + 8.00e3·20-s − 2.72e4·22-s + 1.90e4·23-s − 6.09e4·25-s + 8.13e4·26-s + 1.90e4·28-s + 9.33e3·29-s + 5.64e4·31-s + 1.58e5·32-s + 1.75e5·34-s + 4.08e4·35-s + 5.60e4·37-s + 5.23e4·38-s + 2.02e5·40-s + 5.99e5·41-s + ⋯ |
| L(s) = 1 | + 0.723·2-s − 0.477·4-s − 0.468·5-s − 0.343·7-s − 1.06·8-s − 0.339·10-s − 0.755·11-s + 1.25·13-s − 0.248·14-s − 0.295·16-s + 1.05·17-s + 0.213·19-s + 0.223·20-s − 0.546·22-s + 0.326·23-s − 0.780·25-s + 0.907·26-s + 0.163·28-s + 0.0711·29-s + 0.340·31-s + 0.854·32-s + 0.764·34-s + 0.160·35-s + 0.181·37-s + 0.154·38-s + 0.500·40-s + 1.35·41-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 | \( 1 - 8.18T + 128T^{2} \) |
| 5 | \( 1 + 131.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 311.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 3.33e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 9.94e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 2.14e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 6.39e3T + 8.93e8T^{2} \) |
| 23 | \( 1 - 1.90e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 9.33e3T + 1.72e10T^{2} \) |
| 31 | \( 1 - 5.64e4T + 2.75e10T^{2} \) |
| 37 | \( 1 - 5.60e4T + 9.49e10T^{2} \) |
| 41 | \( 1 - 5.99e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 4.98e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 2.87e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 3.44e5T + 1.17e12T^{2} \) |
| 61 | \( 1 - 8.23e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + 1.16e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 2.86e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 3.54e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 6.70e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 2.90e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 6.75e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.23e7T + 8.07e13T^{2} \) |
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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
−9.252986042954837720610709645685, −8.317817865510172177248001443992, −7.55132438447120673969962933456, −6.17294259050631223873702564990, −5.56253965887304442897477367682, −4.46154948998300005370216080801, −3.60807573233350369575698338023, −2.83486924546413696098378396695, −1.09989516998443313388988440750, 0,
1.09989516998443313388988440750, 2.83486924546413696098378396695, 3.60807573233350369575698338023, 4.46154948998300005370216080801, 5.56253965887304442897477367682, 6.17294259050631223873702564990, 7.55132438447120673969962933456, 8.317817865510172177248001443992, 9.252986042954837720610709645685
