| L(s) = 1 | + 4·2-s + 9·3-s + 16·4-s − 46.9·5-s + 36·6-s + 64·8-s + 81·9-s − 187.·10-s − 87.4·11-s + 144·12-s − 754.·13-s − 422.·15-s + 256·16-s + 1.44e3·17-s + 324·18-s − 2.54e3·19-s − 750.·20-s − 349.·22-s − 912.·23-s + 576·24-s − 922.·25-s − 3.01e3·26-s + 729·27-s + 173.·29-s − 1.68e3·30-s − 4.53e3·31-s + 1.02e3·32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.839·5-s + 0.408·6-s + 0.353·8-s + 0.333·9-s − 0.593·10-s − 0.217·11-s + 0.288·12-s − 1.23·13-s − 0.484·15-s + 0.250·16-s + 1.21·17-s + 0.235·18-s − 1.61·19-s − 0.419·20-s − 0.154·22-s − 0.359·23-s + 0.204·24-s − 0.295·25-s − 0.875·26-s + 0.192·27-s + 0.0382·29-s − 0.342·30-s − 0.846·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 9T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 46.9T + 3.12e3T^{2} \) |
| 11 | \( 1 + 87.4T + 1.61e5T^{2} \) |
| 13 | \( 1 + 754.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.44e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.54e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 912.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 173.T + 2.05e7T^{2} \) |
| 31 | \( 1 + 4.53e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 6.82e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.30e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.22e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.34e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 9.67e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.03e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 732.T + 8.44e8T^{2} \) |
| 67 | \( 1 - 4.63e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 3.29e3T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.02e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 9.85e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 8.77e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 6.90e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 4.21e4T + 8.58e9T^{2} \) |
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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
−10.51590376643020948724991736487, −9.618435821522000277883246917381, −8.222255341193597779137211559764, −7.63949462728791222179988627107, −6.56065676400068965456360293352, −5.17126059653199896645049647197, −4.15823314085556576211956936436, −3.18018357860610324492688567034, −1.95131346687695533031826424056, 0,
1.95131346687695533031826424056, 3.18018357860610324492688567034, 4.15823314085556576211956936436, 5.17126059653199896645049647197, 6.56065676400068965456360293352, 7.63949462728791222179988627107, 8.222255341193597779137211559764, 9.618435821522000277883246917381, 10.51590376643020948724991736487
