| L(s) = 1 | − 8·2-s − 25.5·3-s + 64·4-s + 439.·5-s + 204.·6-s − 363.·7-s − 512·8-s − 1.53e3·9-s − 3.51e3·10-s + 8.20e3·11-s − 1.63e3·12-s − 7.47e3·13-s + 2.91e3·14-s − 1.12e4·15-s + 4.09e3·16-s − 3.15e4·17-s + 1.22e4·18-s + 4.99e4·19-s + 2.81e4·20-s + 9.29e3·21-s − 6.56e4·22-s + 5.76e3·23-s + 1.30e4·24-s + 1.15e5·25-s + 5.98e4·26-s + 9.50e4·27-s − 2.32e4·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.545·3-s + 0.5·4-s + 1.57·5-s + 0.385·6-s − 0.401·7-s − 0.353·8-s − 0.702·9-s − 1.11·10-s + 1.85·11-s − 0.272·12-s − 0.943·13-s + 0.283·14-s − 0.859·15-s + 0.250·16-s − 1.55·17-s + 0.496·18-s + 1.67·19-s + 0.787·20-s + 0.218·21-s − 1.31·22-s + 0.0988·23-s + 0.192·24-s + 1.47·25-s + 0.667·26-s + 0.929·27-s − 0.200·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 8T \) |
| 269 | \( 1 - 1.94e7T \) |
| good | 3 | \( 1 + 25.5T + 2.18e3T^{2} \) |
| 5 | \( 1 - 439.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 363.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 8.20e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 7.47e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 3.15e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 4.99e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 5.76e3T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.47e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.40e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 4.65e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 6.98e4T + 1.94e11T^{2} \) |
| 43 | \( 1 + 2.57e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 8.08e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 5.98e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 1.25e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 1.61e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 2.52e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 1.78e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 5.45e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 4.14e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 1.41e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 1.02e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 7.08e6T + 8.07e13T^{2} \) |
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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
−9.305891543163204427661547000833, −8.826798255620650827029113652472, −7.19246174634155869852945435771, −6.47066773252987709434951656704, −5.85621478342463913805268661568, −4.84172158971979751993903161622, −3.20187107066443377426852700110, −2.09351942524124146508776283799, −1.21493351945459829788566730778, 0,
1.21493351945459829788566730778, 2.09351942524124146508776283799, 3.20187107066443377426852700110, 4.84172158971979751993903161622, 5.85621478342463913805268661568, 6.47066773252987709434951656704, 7.19246174634155869852945435771, 8.826798255620650827029113652472, 9.305891543163204427661547000833
