| L(s) = 1 | + 9.23·2-s + 9·3-s + 53.2·4-s + 89.5·5-s + 83.0·6-s − 121.·7-s + 195.·8-s + 81·9-s + 826.·10-s + 542.·11-s + 478.·12-s − 202.·13-s − 1.12e3·14-s + 806.·15-s + 105.·16-s − 431.·17-s + 747.·18-s + 362.·19-s + 4.76e3·20-s − 1.09e3·21-s + 5.00e3·22-s + 1.69e3·23-s + 1.76e3·24-s + 4.89e3·25-s − 1.86e3·26-s + 729·27-s − 6.46e3·28-s + ⋯ |
| L(s) = 1 | + 1.63·2-s + 0.577·3-s + 1.66·4-s + 1.60·5-s + 0.942·6-s − 0.936·7-s + 1.08·8-s + 0.333·9-s + 2.61·10-s + 1.35·11-s + 0.960·12-s − 0.332·13-s − 1.52·14-s + 0.925·15-s + 0.102·16-s − 0.361·17-s + 0.543·18-s + 0.230·19-s + 2.66·20-s − 0.540·21-s + 2.20·22-s + 0.667·23-s + 0.624·24-s + 1.56·25-s − 0.541·26-s + 0.192·27-s − 1.55·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(7.395285689\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.395285689\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 9T \) |
| 59 | \( 1 + 3.48e3T \) |
| good | 2 | \( 1 - 9.23T + 32T^{2} \) |
| 5 | \( 1 - 89.5T + 3.12e3T^{2} \) |
| 7 | \( 1 + 121.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 542.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 202.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 431.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 362.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 1.69e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 7.67e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 2.42e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 2.53e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 6.56e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 779.T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.38e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.45e4T + 4.18e8T^{2} \) |
| 61 | \( 1 - 1.96e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 5.61e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 3.71e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 8.20e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 7.74e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 6.36e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.55e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 9.30e4T + 8.58e9T^{2} \) |
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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
−12.29203123490503357480936869552, −11.03073798507435375214782482089, −9.605495224231552382208385457342, −9.178038374488707298949832676018, −7.00123663888438529834581255884, −6.28626988683504916246065834658, −5.36657180755255389582708067567, −3.97209537513429716908960495787, −2.87837749579245887162341745406, −1.75789693841219144223883818994,
1.75789693841219144223883818994, 2.87837749579245887162341745406, 3.97209537513429716908960495787, 5.36657180755255389582708067567, 6.28626988683504916246065834658, 7.00123663888438529834581255884, 9.178038374488707298949832676018, 9.605495224231552382208385457342, 11.03073798507435375214782482089, 12.29203123490503357480936869552
