| L(s) = 1 | + 8.52·2-s + 40.6·4-s + 25·5-s + 160.·7-s + 73.3·8-s + 213.·10-s + 279.·11-s − 541.·13-s + 1.36e3·14-s − 674.·16-s + 777.·17-s − 2.68e3·19-s + 1.01e3·20-s + 2.38e3·22-s − 3.69e3·23-s + 625·25-s − 4.61e3·26-s + 6.51e3·28-s + 8.35e3·29-s − 262.·31-s − 8.09e3·32-s + 6.62e3·34-s + 4.01e3·35-s − 1.49e4·37-s − 2.28e4·38-s + 1.83e3·40-s − 7.98e3·41-s + ⋯ |
| L(s) = 1 | + 1.50·2-s + 1.26·4-s + 0.447·5-s + 1.23·7-s + 0.404·8-s + 0.673·10-s + 0.696·11-s − 0.888·13-s + 1.86·14-s − 0.658·16-s + 0.652·17-s − 1.70·19-s + 0.567·20-s + 1.04·22-s − 1.45·23-s + 0.200·25-s − 1.33·26-s + 1.57·28-s + 1.84·29-s − 0.0491·31-s − 1.39·32-s + 0.982·34-s + 0.553·35-s − 1.79·37-s − 2.56·38-s + 0.181·40-s − 0.742·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.837546246\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.837546246\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - 25T \) |
| good | 2 | \( 1 - 8.52T + 32T^{2} \) |
| 7 | \( 1 - 160.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 279.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 541.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 777.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.68e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 3.69e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 8.35e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 262.T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.49e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 7.98e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 5.13e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.05e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.10e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.56e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 8.19e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 5.19e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 2.36e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.09e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 3.92e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 3.59e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 9.40e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.22e4T + 8.58e9T^{2} \) |
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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
−14.37784536888453374327419702333, −14.03683634096722307094210643198, −12.48918967488013551504117405756, −11.77617916094901859396700299413, −10.32377565394036678169206692921, −8.460563802580778649482788226914, −6.66667448731695673263077309964, −5.29318868746665421033732889688, −4.15975404841838252730105105759, −2.13598064014564555158892296757,
2.13598064014564555158892296757, 4.15975404841838252730105105759, 5.29318868746665421033732889688, 6.66667448731695673263077309964, 8.460563802580778649482788226914, 10.32377565394036678169206692921, 11.77617916094901859396700299413, 12.48918967488013551504117405756, 14.03683634096722307094210643198, 14.37784536888453374327419702333
