| L(s) = 1 | + 9.12·2-s + 9·3-s + 51.1·4-s + 25·5-s + 82.0·6-s + 202.·7-s + 175.·8-s + 81·9-s + 228.·10-s + 121·11-s + 460.·12-s − 622.·13-s + 1.84e3·14-s + 225·15-s − 41.4·16-s − 1.58e3·17-s + 738.·18-s + 1.44e3·19-s + 1.27e3·20-s + 1.82e3·21-s + 1.10e3·22-s − 1.55e3·23-s + 1.57e3·24-s + 625·25-s − 5.68e3·26-s + 729·27-s + 1.03e4·28-s + ⋯ |
| L(s) = 1 | + 1.61·2-s + 0.577·3-s + 1.59·4-s + 0.447·5-s + 0.930·6-s + 1.56·7-s + 0.967·8-s + 0.333·9-s + 0.721·10-s + 0.301·11-s + 0.923·12-s − 1.02·13-s + 2.51·14-s + 0.258·15-s − 0.0404·16-s − 1.33·17-s + 0.537·18-s + 0.917·19-s + 0.715·20-s + 0.901·21-s + 0.486·22-s − 0.614·23-s + 0.558·24-s + 0.200·25-s − 1.64·26-s + 0.192·27-s + 2.49·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(7.023699717\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.023699717\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 9T \) |
| 5 | \( 1 - 25T \) |
| 11 | \( 1 - 121T \) |
| good | 2 | \( 1 - 9.12T + 32T^{2} \) |
| 7 | \( 1 - 202.T + 1.68e4T^{2} \) |
| 13 | \( 1 + 622.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.58e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.44e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 1.55e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 4.45e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 5.06e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.16e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 4.84e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.92e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.33e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 4.24e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.32e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.99e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 7.11e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 7.21e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 4.91e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 2.52e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 5.18e3T + 3.93e9T^{2} \) |
| 89 | \( 1 - 7.41e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.55e4T + 8.58e9T^{2} \) |
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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
−11.97035574984152750018029537339, −11.45746293607332836053063931359, −10.08968825173930940609719481334, −8.714997038162604200274856513215, −7.53414924673951534308406837484, −6.34562867134847575648743315765, −4.95576881670111944660011954891, −4.43961935552296804158376560892, −2.81200251609760801624871116811, −1.76456255428300567737879860238,
1.76456255428300567737879860238, 2.81200251609760801624871116811, 4.43961935552296804158376560892, 4.95576881670111944660011954891, 6.34562867134847575648743315765, 7.53414924673951534308406837484, 8.714997038162604200274856513215, 10.08968825173930940609719481334, 11.45746293607332836053063931359, 11.97035574984152750018029537339
