| L(s) = 1 | + 0.471·2-s − 31.7·4-s + 25·5-s − 149.·7-s − 30.0·8-s + 11.7·10-s + 121·11-s + 286.·13-s − 70.4·14-s + 1.00e3·16-s − 245.·17-s + 1.74e3·19-s − 794.·20-s + 57.0·22-s − 2.64e3·23-s + 625·25-s + 135.·26-s + 4.74e3·28-s + 5.49e3·29-s + 5.31e3·31-s + 1.43e3·32-s − 116.·34-s − 3.73e3·35-s − 39.6·37-s + 820.·38-s − 752.·40-s − 6.36e3·41-s + ⋯ |
| L(s) = 1 | + 0.0833·2-s − 0.993·4-s + 0.447·5-s − 1.15·7-s − 0.166·8-s + 0.0372·10-s + 0.301·11-s + 0.470·13-s − 0.0960·14-s + 0.979·16-s − 0.206·17-s + 1.10·19-s − 0.444·20-s + 0.0251·22-s − 1.04·23-s + 0.200·25-s + 0.0392·26-s + 1.14·28-s + 1.21·29-s + 0.993·31-s + 0.247·32-s − 0.0172·34-s − 0.515·35-s − 0.00475·37-s + 0.0922·38-s − 0.0743·40-s − 0.591·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 - 25T \) |
| 11 | \( 1 - 121T \) |
| good | 2 | \( 1 - 0.471T + 32T^{2} \) |
| 7 | \( 1 + 149.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 286.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 245.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.74e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 2.64e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 5.49e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 5.31e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 39.6T + 6.93e7T^{2} \) |
| 41 | \( 1 + 6.36e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.51e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 541.T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.27e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.03e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 7.71e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.71e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 2.31e3T + 1.80e9T^{2} \) |
| 73 | \( 1 + 4.34e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 2.88e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 6.20e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 9.22e3T + 5.58e9T^{2} \) |
| 97 | \( 1 + 2.79e4T + 8.58e9T^{2} \) |
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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
−9.902747227660815265390065471168, −8.902915817721568091654196838431, −8.113617292753896543664305962865, −6.73125656891723333527854350184, −5.98780641352280489139006044776, −4.95019283594582153208465657340, −3.82471156613486901299440523558, −2.92492691892149185637403606229, −1.21169288996050740501531569640, 0,
1.21169288996050740501531569640, 2.92492691892149185637403606229, 3.82471156613486901299440523558, 4.95019283594582153208465657340, 5.98780641352280489139006044776, 6.73125656891723333527854350184, 8.113617292753896543664305962865, 8.902915817721568091654196838431, 9.902747227660815265390065471168
