| L(s) = 1 | − 22·2-s + 356·4-s + 125·5-s − 1.36e3·7-s − 5.01e3·8-s − 2.75e3·10-s − 4.20e3·11-s − 2.19e3·13-s + 3.00e4·14-s + 6.47e4·16-s − 2.23e4·17-s + 3.61e4·19-s + 4.45e4·20-s + 9.24e4·22-s − 4.98e4·23-s + 1.56e4·25-s + 4.83e4·26-s − 4.86e5·28-s − 1.94e5·29-s − 1.46e5·31-s − 7.83e5·32-s + 4.90e5·34-s − 1.70e5·35-s + 3.14e5·37-s − 7.95e5·38-s − 6.27e5·40-s − 2.27e5·41-s + ⋯ |
| L(s) = 1 | − 1.94·2-s + 2.78·4-s + 0.447·5-s − 1.50·7-s − 3.46·8-s − 0.869·10-s − 0.952·11-s − 0.277·13-s + 2.92·14-s + 3.95·16-s − 1.10·17-s + 1.20·19-s + 1.24·20-s + 1.85·22-s − 0.853·23-s + 1/5·25-s + 0.539·26-s − 4.18·28-s − 1.48·29-s − 0.886·31-s − 4.22·32-s + 2.14·34-s − 0.673·35-s + 1.02·37-s − 2.35·38-s − 1.54·40-s − 0.516·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - p^{3} T \) |
| 13 | \( 1 + p^{3} T \) |
| good | 2 | \( 1 + 11 p T + p^{7} T^{2} \) |
| 7 | \( 1 + 1366 T + p^{7} T^{2} \) |
| 11 | \( 1 + 4204 T + p^{7} T^{2} \) |
| 17 | \( 1 + 22300 T + p^{7} T^{2} \) |
| 19 | \( 1 - 1902 p T + p^{7} T^{2} \) |
| 23 | \( 1 + 49830 T + p^{7} T^{2} \) |
| 29 | \( 1 + 194704 T + p^{7} T^{2} \) |
| 31 | \( 1 + 146992 T + p^{7} T^{2} \) |
| 37 | \( 1 - 314494 T + p^{7} T^{2} \) |
| 41 | \( 1 + 227838 T + p^{7} T^{2} \) |
| 43 | \( 1 + 1028276 T + p^{7} T^{2} \) |
| 47 | \( 1 + 612488 T + p^{7} T^{2} \) |
| 53 | \( 1 + 890902 T + p^{7} T^{2} \) |
| 59 | \( 1 - 2169300 T + p^{7} T^{2} \) |
| 61 | \( 1 + 2557954 T + p^{7} T^{2} \) |
| 67 | \( 1 - 748436 T + p^{7} T^{2} \) |
| 71 | \( 1 + 1954176 T + p^{7} T^{2} \) |
| 73 | \( 1 + 3184332 T + p^{7} T^{2} \) |
| 79 | \( 1 - 2813208 T + p^{7} T^{2} \) |
| 83 | \( 1 + 3942316 T + p^{7} T^{2} \) |
| 89 | \( 1 + 1542850 T + p^{7} T^{2} \) |
| 97 | \( 1 + 4724804 T + p^{7} T^{2} \) |
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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
−8.933312486993027875804946103193, −7.916322178235265733407042626914, −7.11671860293146551683596213807, −6.38415562463586559285904729067, −5.50899626922023215978312340979, −3.36494926070622601642304765032, −2.53622373195146212609163872544, −1.58523968224611396264771184227, 0, 0,
1.58523968224611396264771184227, 2.53622373195146212609163872544, 3.36494926070622601642304765032, 5.50899626922023215978312340979, 6.38415562463586559285904729067, 7.11671860293146551683596213807, 7.916322178235265733407042626914, 8.933312486993027875804946103193
