| L(s) = 1 | + 4·2-s − 9·3-s + 16·4-s + 25·5-s − 36·6-s + 208.·7-s + 64·8-s + 81·9-s + 100·10-s + 593.·11-s − 144·12-s − 360.·13-s + 834.·14-s − 225·15-s + 256·16-s + 1.46e3·17-s + 324·18-s + 361·19-s + 400·20-s − 1.87e3·21-s + 2.37e3·22-s + 2.45e3·23-s − 576·24-s + 625·25-s − 1.44e3·26-s − 729·27-s + 3.33e3·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.447·5-s − 0.408·6-s + 1.60·7-s + 0.353·8-s + 0.333·9-s + 0.316·10-s + 1.47·11-s − 0.288·12-s − 0.592·13-s + 1.13·14-s − 0.258·15-s + 0.250·16-s + 1.22·17-s + 0.235·18-s + 0.229·19-s + 0.223·20-s − 0.929·21-s + 1.04·22-s + 0.966·23-s − 0.204·24-s + 0.200·25-s − 0.418·26-s − 0.192·27-s + 0.804·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(4.698113818\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.698113818\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 4T \) |
| 3 | \( 1 + 9T \) |
| 5 | \( 1 - 25T \) |
| 19 | \( 1 - 361T \) |
| good | 7 | \( 1 - 208.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 593.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 360.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.46e3T + 1.41e6T^{2} \) |
| 23 | \( 1 - 2.45e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 6.64e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 9.71e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 7.41e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.21e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 37.3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 5.53e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.36e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 5.29e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.75e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 2.33e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 2.29e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 5.02e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 7.40e3T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.08e5T + 3.93e9T^{2} \) |
| 89 | \( 1 - 4.90e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 9.05e4T + 8.58e9T^{2} \) |
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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
−10.14135142182106919073670147344, −9.156357966198705240064708334893, −7.989793220959789533926283268306, −7.11104595797120726259300767914, −6.14892333610209968857924699723, −5.16424922406840522672301561834, −4.63333172687241583840286345363, −3.38857011565962679632232859864, −1.80939837712439926807758317214, −1.12235645755957671849411365895,
1.12235645755957671849411365895, 1.80939837712439926807758317214, 3.38857011565962679632232859864, 4.63333172687241583840286345363, 5.16424922406840522672301561834, 6.14892333610209968857924699723, 7.11104595797120726259300767914, 7.989793220959789533926283268306, 9.156357966198705240064708334893, 10.14135142182106919073670147344
