| L(s) = 1 | − 18.7·2-s + 222.·4-s − 443.·5-s − 1.69e3·7-s − 1.76e3·8-s + 8.30e3·10-s + 2.76e3·11-s − 1.09e4·13-s + 3.17e4·14-s + 4.61e3·16-s + 1.71e4·17-s − 4.12e4·19-s − 9.87e4·20-s − 5.17e4·22-s − 8.25e4·23-s + 1.18e5·25-s + 2.05e5·26-s − 3.77e5·28-s − 1.03e5·29-s − 1.62e5·31-s + 1.39e5·32-s − 3.21e5·34-s + 7.52e5·35-s + 4.46e5·37-s + 7.72e5·38-s + 7.84e5·40-s − 3.71e5·41-s + ⋯ |
| L(s) = 1 | − 1.65·2-s + 1.73·4-s − 1.58·5-s − 1.86·7-s − 1.22·8-s + 2.62·10-s + 0.626·11-s − 1.38·13-s + 3.09·14-s + 0.281·16-s + 0.847·17-s − 1.38·19-s − 2.75·20-s − 1.03·22-s − 1.41·23-s + 1.52·25-s + 2.28·26-s − 3.24·28-s − 0.784·29-s − 0.977·31-s + 0.754·32-s − 1.40·34-s + 2.96·35-s + 1.45·37-s + 2.28·38-s + 1.93·40-s − 0.841·41-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 | \( 1 + 18.7T + 128T^{2} \) |
| 5 | \( 1 + 443.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 1.69e3T + 8.23e5T^{2} \) |
| 11 | \( 1 - 2.76e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 1.09e4T + 6.27e7T^{2} \) |
| 17 | \( 1 - 1.71e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 4.12e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 8.25e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.03e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.62e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 4.46e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 3.71e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 7.90e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 1.11e6T + 5.06e11T^{2} \) |
| 53 | \( 1 - 7.71e5T + 1.17e12T^{2} \) |
| 61 | \( 1 + 2.11e5T + 3.14e12T^{2} \) |
| 67 | \( 1 - 1.48e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 3.40e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 3.93e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 4.55e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 4.51e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 4.21e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.59e6T + 8.07e13T^{2} \) |
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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
−9.317023405660320512407622534281, −8.429582145196564365307287593334, −7.53504888254196675978847342968, −7.02731938004571121739008792524, −6.09193938800666538743724587039, −4.21130653437871494744377129271, −3.35816252558520548046078484019, −2.18069913315920447863644747827, −0.53674707995036547970847649452, 0,
0.53674707995036547970847649452, 2.18069913315920447863644747827, 3.35816252558520548046078484019, 4.21130653437871494744377129271, 6.09193938800666538743724587039, 7.02731938004571121739008792524, 7.53504888254196675978847342968, 8.429582145196564365307287593334, 9.317023405660320512407622534281
