| L(s) = 1 | + 4·2-s − 9·3-s + 16·4-s − 25·5-s − 36·6-s + 138.·7-s + 64·8-s + 81·9-s − 100·10-s − 242.·11-s − 144·12-s + 181.·13-s + 554.·14-s + 225·15-s + 256·16-s + 1.88e3·17-s + 324·18-s − 361·19-s − 400·20-s − 1.24e3·21-s − 971.·22-s + 242.·23-s − 576·24-s + 625·25-s + 725.·26-s − 729·27-s + 2.21e3·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.447·5-s − 0.408·6-s + 1.06·7-s + 0.353·8-s + 0.333·9-s − 0.316·10-s − 0.605·11-s − 0.288·12-s + 0.297·13-s + 0.755·14-s + 0.258·15-s + 0.250·16-s + 1.58·17-s + 0.235·18-s − 0.229·19-s − 0.223·20-s − 0.617·21-s − 0.428·22-s + 0.0956·23-s − 0.204·24-s + 0.200·25-s + 0.210·26-s − 0.192·27-s + 0.534·28-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
| \(L(3)\) |
\(\approx\) |
\(3.053307616\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.053307616\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 - 138.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 242.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 181.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.88e3T + 1.41e6T^{2} \) |
| 23 | \( 1 - 242.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 7.82e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 5.94e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 7.81e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.14e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.12e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 3.90e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.15e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.72e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.30e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.46e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 1.96e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.69e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 1.18e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 7.42e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 2.36e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 5.12e4T + 8.58e9T^{2} \) |
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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
−10.32024104282372780649757303011, −9.058410799849757266908024578100, −7.66702622173711287897509738278, −7.54757828271915041644968044041, −5.95213873566170704399921710878, −5.37299783294497656264493394967, −4.41788926769184153985921463998, −3.45534159594730420310606708601, −2.01610263804630265029165995621, −0.814487384494860875322207991801,
0.814487384494860875322207991801, 2.01610263804630265029165995621, 3.45534159594730420310606708601, 4.41788926769184153985921463998, 5.37299783294497656264493394967, 5.95213873566170704399921710878, 7.54757828271915041644968044041, 7.66702622173711287897509738278, 9.058410799849757266908024578100, 10.32024104282372780649757303011
