| L(s) = 1 | + 10.3·2-s − 10.7·3-s + 74.7·4-s − 25·5-s − 111.·6-s + 442.·8-s − 126.·9-s − 258.·10-s − 262.·11-s − 806.·12-s − 688.·13-s + 269.·15-s + 2.17e3·16-s − 1.67e3·17-s − 1.31e3·18-s − 925.·19-s − 1.86e3·20-s − 2.71e3·22-s + 4.60e3·23-s − 4.76e3·24-s + 625·25-s − 7.11e3·26-s + 3.98e3·27-s − 3.41e3·29-s + 2.78e3·30-s + 5.53e3·31-s + 8.34e3·32-s + ⋯ |
| L(s) = 1 | + 1.82·2-s − 0.691·3-s + 2.33·4-s − 0.447·5-s − 1.26·6-s + 2.44·8-s − 0.521·9-s − 0.816·10-s − 0.655·11-s − 1.61·12-s − 1.13·13-s + 0.309·15-s + 2.12·16-s − 1.40·17-s − 0.953·18-s − 0.587·19-s − 1.04·20-s − 1.19·22-s + 1.81·23-s − 1.68·24-s + 0.200·25-s − 2.06·26-s + 1.05·27-s − 0.753·29-s + 0.564·30-s + 1.03·31-s + 1.44·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 25T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 10.3T + 32T^{2} \) |
| 3 | \( 1 + 10.7T + 243T^{2} \) |
| 11 | \( 1 + 262.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 688.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.67e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 925.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 4.60e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 3.41e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 5.53e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.49e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.49e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.96e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.12e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.13e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.47e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.23e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 4.49e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 1.09e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 9.20e3T + 2.07e9T^{2} \) |
| 79 | \( 1 + 3.35e3T + 3.07e9T^{2} \) |
| 83 | \( 1 + 4.55e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 5.28e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.64e5T + 8.58e9T^{2} \) |
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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
−11.21549095824338804720739150306, −10.41957847493319476436960878432, −8.599613728069086516030396174759, −7.12626539862198594140891431569, −6.49284686452283808900517150745, −5.11733264437658635675443882789, −4.83567140378609100515879814365, −3.34017073998184507281108673101, −2.27312931224302467961178707324, 0,
2.27312931224302467961178707324, 3.34017073998184507281108673101, 4.83567140378609100515879814365, 5.11733264437658635675443882789, 6.49284686452283808900517150745, 7.12626539862198594140891431569, 8.599613728069086516030396174759, 10.41957847493319476436960878432, 11.21549095824338804720739150306
