| L(s) = 1 | + 4·2-s − 9·3-s + 16·4-s + 36.7·5-s − 36·6-s − 14.5·7-s + 64·8-s + 81·9-s + 146.·10-s − 248.·11-s − 144·12-s − 302.·13-s − 58.0·14-s − 330.·15-s + 256·16-s − 1.83e3·17-s + 324·18-s + 1.96e3·19-s + 587.·20-s + 130.·21-s − 992.·22-s − 2.05e3·23-s − 576·24-s − 1.77e3·25-s − 1.21e3·26-s − 729·27-s − 232.·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.656·5-s − 0.408·6-s − 0.111·7-s + 0.353·8-s + 0.333·9-s + 0.464·10-s − 0.618·11-s − 0.288·12-s − 0.496·13-s − 0.0791·14-s − 0.379·15-s + 0.250·16-s − 1.53·17-s + 0.235·18-s + 1.25·19-s + 0.328·20-s + 0.0646·21-s − 0.437·22-s − 0.809·23-s − 0.204·24-s − 0.568·25-s − 0.351·26-s − 0.192·27-s − 0.0559·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + 9T \) |
| 59 | \( 1 - 3.48e3T \) |
| good | 5 | \( 1 - 36.7T + 3.12e3T^{2} \) |
| 7 | \( 1 + 14.5T + 1.68e4T^{2} \) |
| 11 | \( 1 + 248.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 302.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.83e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.96e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 2.05e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 847.T + 2.05e7T^{2} \) |
| 31 | \( 1 - 2.38e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 6.70e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 2.46e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.71e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.71e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.43e4T + 4.18e8T^{2} \) |
| 61 | \( 1 + 2.99e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 2.15e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 6.18e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.55e3T + 2.07e9T^{2} \) |
| 79 | \( 1 + 8.05e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 2.36e3T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.15e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.29e5T + 8.58e9T^{2} \) |
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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
−10.21411976389359520024055387769, −9.586347926413273211107867866726, −8.135755282221432418354909689865, −7.01030857428184845413875949429, −6.13611379457399218019477474317, −5.25933391530630802449344306331, −4.36743517595056870451277087186, −2.87059703519515987314032787763, −1.72546499315015138585326946200, 0,
1.72546499315015138585326946200, 2.87059703519515987314032787763, 4.36743517595056870451277087186, 5.25933391530630802449344306331, 6.13611379457399218019477474317, 7.01030857428184845413875949429, 8.135755282221432418354909689865, 9.586347926413273211107867866726, 10.21411976389359520024055387769
