| L(s) = 1 | + 3.85·2-s + 6.86·4-s + 19.5·5-s − 6.26·7-s − 4.36·8-s + 75.3·10-s + 27.2·11-s − 13·13-s − 24.1·14-s − 71.7·16-s − 30.8·17-s + 127.·19-s + 134.·20-s + 105.·22-s − 84.3·23-s + 256.·25-s − 50.1·26-s − 43.0·28-s − 272.·29-s − 166.·31-s − 241.·32-s − 118.·34-s − 122.·35-s − 198.·37-s + 492.·38-s − 85.2·40-s − 160.·41-s + ⋯ |
| L(s) = 1 | + 1.36·2-s + 0.858·4-s + 1.74·5-s − 0.338·7-s − 0.192·8-s + 2.38·10-s + 0.746·11-s − 0.277·13-s − 0.461·14-s − 1.12·16-s − 0.440·17-s + 1.54·19-s + 1.49·20-s + 1.01·22-s − 0.764·23-s + 2.05·25-s − 0.378·26-s − 0.290·28-s − 1.74·29-s − 0.963·31-s − 1.33·32-s − 0.599·34-s − 0.590·35-s − 0.880·37-s + 2.10·38-s − 0.337·40-s − 0.610·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.691596179\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.691596179\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 13 | \( 1 + 13T \) |
| good | 2 | \( 1 - 3.85T + 8T^{2} \) |
| 5 | \( 1 - 19.5T + 125T^{2} \) |
| 7 | \( 1 + 6.26T + 343T^{2} \) |
| 11 | \( 1 - 27.2T + 1.33e3T^{2} \) |
| 17 | \( 1 + 30.8T + 4.91e3T^{2} \) |
| 19 | \( 1 - 127.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 84.3T + 1.21e4T^{2} \) |
| 29 | \( 1 + 272.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 166.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 198.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 160.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 158.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 305.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 356.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 470.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 171.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 1.02e3T + 3.00e5T^{2} \) |
| 71 | \( 1 - 188.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 959.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.03e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 105.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 649.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 707.T + 9.12e5T^{2} \) |
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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
−13.23894210676702086389475740310, −12.43324953562403679385168484462, −11.24868814393677365803000595641, −9.763342925438282699273384608846, −9.157802729041155800550253748225, −6.96138103094564345573001950821, −5.92184992788502698800587133080, −5.18963424150428009812672372367, −3.54762667186843731435185883866, −2.02992181189936806077436563041,
2.02992181189936806077436563041, 3.54762667186843731435185883866, 5.18963424150428009812672372367, 5.92184992788502698800587133080, 6.96138103094564345573001950821, 9.157802729041155800550253748225, 9.763342925438282699273384608846, 11.24868814393677365803000595641, 12.43324953562403679385168484462, 13.23894210676702086389475740310
