| L(s) = 1 | + 0.219·2-s − 1.60·3-s − 1.95·4-s + 5-s − 0.351·6-s + 0.332·7-s − 0.868·8-s − 0.439·9-s + 0.219·10-s + 5.37·11-s + 3.12·12-s + 0.0729·14-s − 1.60·15-s + 3.71·16-s − 5.06·17-s − 0.0965·18-s − 2.26·19-s − 1.95·20-s − 0.531·21-s + 1.18·22-s − 2.83·23-s + 1.38·24-s + 25-s + 5.50·27-s − 0.648·28-s − 2.90·29-s − 0.351·30-s + ⋯ |
| L(s) = 1 | + 0.155·2-s − 0.923·3-s − 0.975·4-s + 0.447·5-s − 0.143·6-s + 0.125·7-s − 0.306·8-s − 0.146·9-s + 0.0694·10-s + 1.61·11-s + 0.901·12-s + 0.0195·14-s − 0.413·15-s + 0.928·16-s − 1.22·17-s − 0.0227·18-s − 0.520·19-s − 0.436·20-s − 0.116·21-s + 0.251·22-s − 0.592·23-s + 0.283·24-s + 0.200·25-s + 1.05·27-s − 0.122·28-s − 0.539·29-s − 0.0641·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 0.219T + 2T^{2} \) |
| 3 | \( 1 + 1.60T + 3T^{2} \) |
| 7 | \( 1 - 0.332T + 7T^{2} \) |
| 11 | \( 1 - 5.37T + 11T^{2} \) |
| 17 | \( 1 + 5.06T + 17T^{2} \) |
| 19 | \( 1 + 2.26T + 19T^{2} \) |
| 23 | \( 1 + 2.83T + 23T^{2} \) |
| 29 | \( 1 + 2.90T + 29T^{2} \) |
| 31 | \( 1 + 5.46T + 31T^{2} \) |
| 37 | \( 1 + 5.97T + 37T^{2} \) |
| 41 | \( 1 + 3.73T + 41T^{2} \) |
| 43 | \( 1 + 5.06T + 43T^{2} \) |
| 47 | \( 1 + 8.34T + 47T^{2} \) |
| 53 | \( 1 + 1.56T + 53T^{2} \) |
| 59 | \( 1 + 2.70T + 59T^{2} \) |
| 61 | \( 1 - 14.1T + 61T^{2} \) |
| 67 | \( 1 + 10.3T + 67T^{2} \) |
| 71 | \( 1 + 12.7T + 71T^{2} \) |
| 73 | \( 1 - 9.68T + 73T^{2} \) |
| 79 | \( 1 - 4.51T + 79T^{2} \) |
| 83 | \( 1 - 4.26T + 83T^{2} \) |
| 89 | \( 1 + 3.22T + 89T^{2} \) |
| 97 | \( 1 - 2.50T + 97T^{2} \) |
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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
−9.699822472341463335414231501990, −8.983927218001932718593169218371, −8.356838763558682635172396165425, −6.81106234299633048910007147564, −6.22060638463534974274976473325, −5.31096956944644217151831540906, −4.48961647964341465753906594015, −3.54431075326241503450763035360, −1.68341924661537648649956902839, 0,
1.68341924661537648649956902839, 3.54431075326241503450763035360, 4.48961647964341465753906594015, 5.31096956944644217151831540906, 6.22060638463534974274976473325, 6.81106234299633048910007147564, 8.356838763558682635172396165425, 8.983927218001932718593169218371, 9.699822472341463335414231501990
