| L(s) = 1 | − 0.114·2-s + 8.15·3-s − 31.9·4-s + 25·5-s − 0.930·6-s + 179.·7-s + 7.30·8-s − 176.·9-s − 2.85·10-s − 121·11-s − 260.·12-s + 748.·13-s − 20.4·14-s + 203.·15-s + 1.02e3·16-s − 328.·17-s + 20.1·18-s + 361·19-s − 799.·20-s + 1.46e3·21-s + 13.8·22-s − 4.10e3·23-s + 59.5·24-s + 625·25-s − 85.4·26-s − 3.42e3·27-s − 5.74e3·28-s + ⋯ |
| L(s) = 1 | − 0.0201·2-s + 0.523·3-s − 0.999·4-s + 0.447·5-s − 0.0105·6-s + 1.38·7-s + 0.0403·8-s − 0.726·9-s − 0.00902·10-s − 0.301·11-s − 0.522·12-s + 1.22·13-s − 0.0279·14-s + 0.233·15-s + 0.998·16-s − 0.275·17-s + 0.0146·18-s + 0.229·19-s − 0.447·20-s + 0.724·21-s + 0.00608·22-s − 1.61·23-s + 0.0211·24-s + 0.200·25-s − 0.0248·26-s − 0.902·27-s − 1.38·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - 25T \) |
| 11 | \( 1 + 121T \) |
| 19 | \( 1 - 361T \) |
| good | 2 | \( 1 + 0.114T + 32T^{2} \) |
| 3 | \( 1 - 8.15T + 243T^{2} \) |
| 7 | \( 1 - 179.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 748.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 328.T + 1.41e6T^{2} \) |
| 23 | \( 1 + 4.10e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 1.14e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 7.45e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.05e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 8.61e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 135.T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.28e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.82e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.03e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.23e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.80e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 4.43e3T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.40e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 2.80e3T + 3.07e9T^{2} \) |
| 83 | \( 1 + 8.57e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 5.80e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 5.54e4T + 8.58e9T^{2} \) |
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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
−8.651231909236960647885545172656, −8.257908081545357416323825936213, −7.41976522817102690760790573143, −5.81951427578484079221126821319, −5.45974627096147259237197593591, −4.30507363328013898423780444533, −3.56438134514950822539666962718, −2.21604661070545889651115011613, −1.32424394861109094283734390531, 0,
1.32424394861109094283734390531, 2.21604661070545889651115011613, 3.56438134514950822539666962718, 4.30507363328013898423780444533, 5.45974627096147259237197593591, 5.81951427578484079221126821319, 7.41976522817102690760790573143, 8.257908081545357416323825936213, 8.651231909236960647885545172656
