| L(s) = 1 | − 3.74·2-s + 13.5·3-s − 17.9·4-s − 25·5-s − 50.6·6-s + 134.·7-s + 187.·8-s − 59.8·9-s + 93.5·10-s + 121·11-s − 243.·12-s + 386.·13-s − 504.·14-s − 338.·15-s − 124.·16-s + 2.02e3·17-s + 223.·18-s + 486.·19-s + 449.·20-s + 1.82e3·21-s − 452.·22-s + 4.78e3·23-s + 2.53e3·24-s + 625·25-s − 1.44e3·26-s − 4.09e3·27-s − 2.42e3·28-s + ⋯ |
| L(s) = 1 | − 0.661·2-s + 0.868·3-s − 0.562·4-s − 0.447·5-s − 0.574·6-s + 1.03·7-s + 1.03·8-s − 0.246·9-s + 0.295·10-s + 0.301·11-s − 0.488·12-s + 0.634·13-s − 0.687·14-s − 0.388·15-s − 0.121·16-s + 1.70·17-s + 0.162·18-s + 0.309·19-s + 0.251·20-s + 0.902·21-s − 0.199·22-s + 1.88·23-s + 0.897·24-s + 0.200·25-s − 0.419·26-s − 1.08·27-s − 0.584·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.462014667\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.462014667\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 25T \) |
| 11 | \( 1 - 121T \) |
| good | 2 | \( 1 + 3.74T + 32T^{2} \) |
| 3 | \( 1 - 13.5T + 243T^{2} \) |
| 7 | \( 1 - 134.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 386.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 2.02e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 486.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 4.78e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 5.40e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 1.03e4T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.53e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 5.35e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.28e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.64e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 947.T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.04e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.18e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.38e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 2.46e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 7.42e3T + 2.07e9T^{2} \) |
| 79 | \( 1 - 6.73e3T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.02e5T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.41e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 4.19e4T + 8.58e9T^{2} \) |
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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
−14.30716047757108380500227401766, −13.50117971011798845824042187233, −11.88093342361022920910353239985, −10.63099511685733417447466493438, −9.203120664614510611354125123630, −8.371968956774877811136417457034, −7.55783307831176245375942467026, −5.11994649061266818245715194219, −3.47888678192176811085009648830, −1.20413217100725281316999819610,
1.20413217100725281316999819610, 3.47888678192176811085009648830, 5.11994649061266818245715194219, 7.55783307831176245375942467026, 8.371968956774877811136417457034, 9.203120664614510611354125123630, 10.63099511685733417447466493438, 11.88093342361022920910353239985, 13.50117971011798845824042187233, 14.30716047757108380500227401766
