| L(s) = 1 | + 8.27·2-s − 59.4·4-s − 230.·5-s + 754.·7-s − 1.55e3·8-s − 1.90e3·10-s + 4.11e3·11-s + 8.27e3·13-s + 6.24e3·14-s − 5.23e3·16-s − 3.30e4·17-s + 4.85e4·19-s + 1.36e4·20-s + 3.40e4·22-s + 1.21e4·23-s − 2.50e4·25-s + 6.85e4·26-s − 4.48e4·28-s − 7.52e4·29-s − 2.32e5·31-s + 1.55e5·32-s − 2.73e5·34-s − 1.73e5·35-s − 2.85e3·37-s + 4.01e5·38-s + 3.57e5·40-s − 1.38e4·41-s + ⋯ |
| L(s) = 1 | + 0.731·2-s − 0.464·4-s − 0.824·5-s + 0.831·7-s − 1.07·8-s − 0.603·10-s + 0.932·11-s + 1.04·13-s + 0.608·14-s − 0.319·16-s − 1.62·17-s + 1.62·19-s + 0.382·20-s + 0.682·22-s + 0.208·23-s − 0.320·25-s + 0.764·26-s − 0.386·28-s − 0.572·29-s − 1.40·31-s + 0.837·32-s − 1.19·34-s − 0.685·35-s − 0.00927·37-s + 1.18·38-s + 0.883·40-s − 0.0314·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 23 | \( 1 - 1.21e4T \) |
| good | 2 | \( 1 - 8.27T + 128T^{2} \) |
| 5 | \( 1 + 230.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 754.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 4.11e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 8.27e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 3.30e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 4.85e4T + 8.93e8T^{2} \) |
| 29 | \( 1 + 7.52e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 2.32e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 2.85e3T + 9.49e10T^{2} \) |
| 41 | \( 1 + 1.38e4T + 1.94e11T^{2} \) |
| 43 | \( 1 + 3.31e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 1.82e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 9.43e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 1.66e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 2.55e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 1.83e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 2.39e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 6.34e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 1.14e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 9.04e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 6.35e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.01e7T + 8.07e13T^{2} \) |
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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
−11.19717773633902451911165978039, −9.389623769691042146098470781201, −8.690010491887983968571971087258, −7.61316533497305954812659457603, −6.32637415973230480283708585443, −5.11343756053282229591081369592, −4.15619131545339255861293349584, −3.38788121766483624288513969118, −1.48083714603107914121620520561, 0,
1.48083714603107914121620520561, 3.38788121766483624288513969118, 4.15619131545339255861293349584, 5.11343756053282229591081369592, 6.32637415973230480283708585443, 7.61316533497305954812659457603, 8.690010491887983968571971087258, 9.389623769691042146098470781201, 11.19717773633902451911165978039
