| L(s) = 1 | + 3.70·2-s − 18.2·4-s + 28.0·5-s − 2.54·7-s − 186.·8-s + 103.·10-s + 419.·11-s − 176.·13-s − 9.44·14-s − 107.·16-s + 688.·17-s − 3.05e3·19-s − 510.·20-s + 1.55e3·22-s − 529·23-s − 2.34e3·25-s − 654.·26-s + 46.4·28-s − 7.24e3·29-s − 2.82e3·31-s + 5.56e3·32-s + 2.55e3·34-s − 71.2·35-s + 1.22e4·37-s − 1.13e4·38-s − 5.21e3·40-s − 1.65e4·41-s + ⋯ |
| L(s) = 1 | + 0.655·2-s − 0.570·4-s + 0.501·5-s − 0.0196·7-s − 1.02·8-s + 0.328·10-s + 1.04·11-s − 0.289·13-s − 0.0128·14-s − 0.104·16-s + 0.578·17-s − 1.94·19-s − 0.285·20-s + 0.685·22-s − 0.208·23-s − 0.748·25-s − 0.189·26-s + 0.0111·28-s − 1.59·29-s − 0.527·31-s + 0.960·32-s + 0.379·34-s − 0.00983·35-s + 1.47·37-s − 1.27·38-s − 0.515·40-s − 1.53·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 23 | \( 1 + 529T \) |
| good | 2 | \( 1 - 3.70T + 32T^{2} \) |
| 5 | \( 1 - 28.0T + 3.12e3T^{2} \) |
| 7 | \( 1 + 2.54T + 1.68e4T^{2} \) |
| 11 | \( 1 - 419.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 176.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 688.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 3.05e3T + 2.47e6T^{2} \) |
| 29 | \( 1 + 7.24e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 2.82e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.22e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.65e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.32e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 9.20e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.03e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.51e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.80e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 6.40e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 5.70e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 7.42e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 3.83e3T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.56e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 3.17e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 741.T + 8.58e9T^{2} \) |
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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
−11.19417142994724126502000656562, −9.864308237258883517933884768389, −9.187300669602672094180871699893, −8.105135372682296749127531928503, −6.54636951976347425866789114500, −5.72311011927678266516074318050, −4.49293360953195282514851332709, −3.53179233591950710068510942755, −1.85823297914250556078489329169, 0,
1.85823297914250556078489329169, 3.53179233591950710068510942755, 4.49293360953195282514851332709, 5.72311011927678266516074318050, 6.54636951976347425866789114500, 8.105135372682296749127531928503, 9.187300669602672094180871699893, 9.864308237258883517933884768389, 11.19417142994724126502000656562
