| L(s) = 1 | − 2.64·2-s − 0.999·4-s + 10.5·5-s − 22·7-s + 23.8·8-s − 28.0·10-s + 5.29·11-s + 13·13-s + 58.2·14-s − 55.0·16-s − 116.·17-s − 126·19-s − 10.5·20-s − 14.0·22-s − 31.7·23-s − 12.9·25-s − 34.3·26-s + 21.9·28-s − 52.9·29-s − 182·31-s − 44.9·32-s + 308·34-s − 232.·35-s − 86·37-s + 333.·38-s + 252.·40-s + 444.·41-s + ⋯ |
| L(s) = 1 | − 0.935·2-s − 0.124·4-s + 0.946·5-s − 1.18·7-s + 1.05·8-s − 0.885·10-s + 0.145·11-s + 0.277·13-s + 1.11·14-s − 0.859·16-s − 1.66·17-s − 1.52·19-s − 0.118·20-s − 0.135·22-s − 0.287·23-s − 0.103·25-s − 0.259·26-s + 0.148·28-s − 0.338·29-s − 1.05·31-s − 0.248·32-s + 1.55·34-s − 1.12·35-s − 0.382·37-s + 1.42·38-s + 0.996·40-s + 1.69·41-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
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 2.64T + 8T^{2} \) |
| 5 | \( 1 - 10.5T + 125T^{2} \) |
| 7 | \( 1 + 22T + 343T^{2} \) |
| 11 | \( 1 - 5.29T + 1.33e3T^{2} \) |
| 17 | \( 1 + 116.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 126T + 6.85e3T^{2} \) |
| 23 | \( 1 + 31.7T + 1.21e4T^{2} \) |
| 29 | \( 1 + 52.9T + 2.43e4T^{2} \) |
| 31 | \( 1 + 182T + 2.97e4T^{2} \) |
| 37 | \( 1 + 86T + 5.06e4T^{2} \) |
| 41 | \( 1 - 444.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 96T + 7.95e4T^{2} \) |
| 47 | \( 1 + 365.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 190.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 587.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 574T + 2.26e5T^{2} \) |
| 67 | \( 1 + 530T + 3.00e5T^{2} \) |
| 71 | \( 1 + 809.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 154T + 3.89e5T^{2} \) |
| 79 | \( 1 + 460T + 4.93e5T^{2} \) |
| 83 | \( 1 - 322.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.43e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 70T + 9.12e5T^{2} \) |
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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
−12.99150470352574584035942427316, −11.06948416915141318041468229486, −10.15143470278430850365840553124, −9.304977165246293260978478938253, −8.611307942305103291324962083543, −6.98086670748887724328104797280, −5.97055871662157869782917407107, −4.17973981511170068750084622342, −2.08744078499639421461408287849, 0,
2.08744078499639421461408287849, 4.17973981511170068750084622342, 5.97055871662157869782917407107, 6.98086670748887724328104797280, 8.611307942305103291324962083543, 9.304977165246293260978478938253, 10.15143470278430850365840553124, 11.06948416915141318041468229486, 12.99150470352574584035942427316
