| L(s) = 1 | − 4·2-s − 15·3-s − 16·4-s − 19·5-s + 60·6-s + 10·7-s + 192·8-s − 18·9-s + 76·10-s − 121·11-s + 240·12-s − 1.14e3·13-s − 40·14-s + 285·15-s − 256·16-s + 686·17-s + 72·18-s − 384·19-s + 304·20-s − 150·21-s + 484·22-s + 3.70e3·23-s − 2.88e3·24-s − 2.76e3·25-s + 4.59e3·26-s + 3.91e3·27-s − 160·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.962·3-s − 1/2·4-s − 0.339·5-s + 0.680·6-s + 0.0771·7-s + 1.06·8-s − 0.0740·9-s + 0.240·10-s − 0.301·11-s + 0.481·12-s − 1.88·13-s − 0.0545·14-s + 0.327·15-s − 1/4·16-s + 0.575·17-s + 0.0523·18-s − 0.244·19-s + 0.169·20-s − 0.0742·21-s + 0.213·22-s + 1.46·23-s − 1.02·24-s − 0.884·25-s + 1.33·26-s + 1.03·27-s − 0.0385·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + p^{2} T \) |
| good | 2 | \( 1 + p^{2} T + p^{5} T^{2} \) |
| 3 | \( 1 + 5 p T + p^{5} T^{2} \) |
| 5 | \( 1 + 19 T + p^{5} T^{2} \) |
| 7 | \( 1 - 10 T + p^{5} T^{2} \) |
| 13 | \( 1 + 1148 T + p^{5} T^{2} \) |
| 17 | \( 1 - 686 T + p^{5} T^{2} \) |
| 19 | \( 1 + 384 T + p^{5} T^{2} \) |
| 23 | \( 1 - 3709 T + p^{5} T^{2} \) |
| 29 | \( 1 + 5424 T + p^{5} T^{2} \) |
| 31 | \( 1 + 6443 T + p^{5} T^{2} \) |
| 37 | \( 1 - 12063 T + p^{5} T^{2} \) |
| 41 | \( 1 + 1528 T + p^{5} T^{2} \) |
| 43 | \( 1 + 4026 T + p^{5} T^{2} \) |
| 47 | \( 1 - 7168 T + p^{5} T^{2} \) |
| 53 | \( 1 + 29862 T + p^{5} T^{2} \) |
| 59 | \( 1 + 6461 T + p^{5} T^{2} \) |
| 61 | \( 1 + 16980 T + p^{5} T^{2} \) |
| 67 | \( 1 - 29999 T + p^{5} T^{2} \) |
| 71 | \( 1 - 31023 T + p^{5} T^{2} \) |
| 73 | \( 1 - 1924 T + p^{5} T^{2} \) |
| 79 | \( 1 - 65138 T + p^{5} T^{2} \) |
| 83 | \( 1 + 102714 T + p^{5} T^{2} \) |
| 89 | \( 1 - 17415 T + p^{5} T^{2} \) |
| 97 | \( 1 - 66905 T + p^{5} T^{2} \) |
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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
−18.66453917350006631020156598224, −17.34118692589075920999624306041, −16.69913667256964641203681911756, −14.67308249913533907952523638524, −12.68629367565083048861198061223, −11.12080365698102680150012896668, −9.571802060934438424890481187086, −7.60848642897152821944170108350, −5.08027800707969298807961174136, 0,
5.08027800707969298807961174136, 7.60848642897152821944170108350, 9.571802060934438424890481187086, 11.12080365698102680150012896668, 12.68629367565083048861198061223, 14.67308249913533907952523638524, 16.69913667256964641203681911756, 17.34118692589075920999624306041, 18.66453917350006631020156598224
