| L(s) = 1 | + 8.50·2-s − 9·3-s + 40.3·4-s − 86.6·5-s − 76.5·6-s − 64.0·7-s + 71.0·8-s + 81·9-s − 737.·10-s − 285.·11-s − 363.·12-s − 307.·13-s − 544.·14-s + 780.·15-s − 686.·16-s + 2.22e3·17-s + 688.·18-s − 1.80e3·19-s − 3.49e3·20-s + 576.·21-s − 2.42e3·22-s − 529·23-s − 639.·24-s + 4.39e3·25-s − 2.61e3·26-s − 729·27-s − 2.58e3·28-s + ⋯ |
| L(s) = 1 | + 1.50·2-s − 0.577·3-s + 1.26·4-s − 1.55·5-s − 0.868·6-s − 0.493·7-s + 0.392·8-s + 0.333·9-s − 2.33·10-s − 0.710·11-s − 0.728·12-s − 0.504·13-s − 0.742·14-s + 0.895·15-s − 0.670·16-s + 1.86·17-s + 0.501·18-s − 1.14·19-s − 1.95·20-s + 0.285·21-s − 1.06·22-s − 0.208·23-s − 0.226·24-s + 1.40·25-s − 0.758·26-s − 0.192·27-s − 0.622·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 9T \) |
| 23 | \( 1 + 529T \) |
| good | 2 | \( 1 - 8.50T + 32T^{2} \) |
| 5 | \( 1 + 86.6T + 3.12e3T^{2} \) |
| 7 | \( 1 + 64.0T + 1.68e4T^{2} \) |
| 11 | \( 1 + 285.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 307.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 2.22e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.80e3T + 2.47e6T^{2} \) |
| 29 | \( 1 + 2.35e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 8.31e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 9.07e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.54e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.53e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.47e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.41e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 8.40e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.61e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.30e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 5.24e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.69e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 1.00e5T + 3.07e9T^{2} \) |
| 83 | \( 1 + 8.52e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 8.30e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 3.17e4T + 8.58e9T^{2} \) |
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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
−12.91117424634476762226056701734, −12.24970046483089594547290846627, −11.53588767485559461505181124363, −10.22248893734734714487959629108, −8.082096600705339305343659983782, −6.85541310429599873619357772477, −5.45138956800723839043256284125, −4.27946264448773410464809790628, −3.14802308510360162417287038197, 0,
3.14802308510360162417287038197, 4.27946264448773410464809790628, 5.45138956800723839043256284125, 6.85541310429599873619357772477, 8.082096600705339305343659983782, 10.22248893734734714487959629108, 11.53588767485559461505181124363, 12.24970046483089594547290846627, 12.91117424634476762226056701734
