| L(s) = 1 | − 32·2-s − 243·3-s + 1.02e3·4-s + 3.12e3·5-s + 7.77e3·6-s − 5.73e4·7-s − 3.27e4·8-s + 5.90e4·9-s − 1.00e5·10-s + 9.54e5·11-s − 2.48e5·12-s + 9.78e5·13-s + 1.83e6·14-s − 7.59e5·15-s + 1.04e6·16-s − 4.57e6·17-s − 1.88e6·18-s + 4.29e5·19-s + 3.20e6·20-s + 1.39e7·21-s − 3.05e7·22-s − 2.56e7·23-s + 7.96e6·24-s + 9.76e6·25-s − 3.12e7·26-s − 1.43e7·27-s − 5.87e7·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 1.29·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 1.78·11-s − 0.288·12-s + 0.730·13-s + 0.912·14-s − 0.258·15-s + 1/4·16-s − 0.781·17-s − 0.235·18-s + 0.0397·19-s + 0.223·20-s + 0.744·21-s − 1.26·22-s − 0.830·23-s + 0.204·24-s + 1/5·25-s − 0.516·26-s − 0.192·27-s − 0.645·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p^{5} T \) |
| 3 | \( 1 + p^{5} T \) |
| 5 | \( 1 - p^{5} T \) |
| good | 7 | \( 1 + 57376 T + p^{11} T^{2} \) |
| 11 | \( 1 - 954372 T + p^{11} T^{2} \) |
| 13 | \( 1 - 978038 T + p^{11} T^{2} \) |
| 17 | \( 1 + 4574766 T + p^{11} T^{2} \) |
| 19 | \( 1 - 429260 T + p^{11} T^{2} \) |
| 23 | \( 1 + 25641792 T + p^{11} T^{2} \) |
| 29 | \( 1 + 188685210 T + p^{11} T^{2} \) |
| 31 | \( 1 - 34469072 T + p^{11} T^{2} \) |
| 37 | \( 1 + 381698146 T + p^{11} T^{2} \) |
| 41 | \( 1 + 1116342918 T + p^{11} T^{2} \) |
| 43 | \( 1 + 182578612 T + p^{11} T^{2} \) |
| 47 | \( 1 - 2055898584 T + p^{11} T^{2} \) |
| 53 | \( 1 + 5352288402 T + p^{11} T^{2} \) |
| 59 | \( 1 + 2306052060 T + p^{11} T^{2} \) |
| 61 | \( 1 - 2262182822 T + p^{11} T^{2} \) |
| 67 | \( 1 + 16091830396 T + p^{11} T^{2} \) |
| 71 | \( 1 - 7283041032 T + p^{11} T^{2} \) |
| 73 | \( 1 - 28423422458 T + p^{11} T^{2} \) |
| 79 | \( 1 + 385693360 T + p^{11} T^{2} \) |
| 83 | \( 1 + 14785428252 T + p^{11} T^{2} \) |
| 89 | \( 1 + 95789444790 T + p^{11} T^{2} \) |
| 97 | \( 1 + 150483759166 T + p^{11} T^{2} \) |
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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
−13.75753066339244915485050051007, −12.41010179387675815620989889614, −11.19842897715948920750099335895, −9.826803515340305548057979496898, −8.949037083018968639024858249023, −6.82234060395048640469383935904, −6.06497768250926829461156042354, −3.69848052712207192955951082405, −1.58480637503248373567245710469, 0,
1.58480637503248373567245710469, 3.69848052712207192955951082405, 6.06497768250926829461156042354, 6.82234060395048640469383935904, 8.949037083018968639024858249023, 9.826803515340305548057979496898, 11.19842897715948920750099335895, 12.41010179387675815620989889614, 13.75753066339244915485050051007
