| L(s) = 1 | + 5-s − 7-s − 2.85·11-s + 5.47·13-s + 3.61·17-s − 19-s − 2.23·23-s − 4·25-s + 6.09·29-s − 10.0·31-s − 35-s − 9.47·37-s + 0.381·41-s − 2·43-s + 0.708·47-s + 49-s + 11.0·53-s − 2.85·55-s − 7.94·59-s − 5·61-s + 5.47·65-s − 15.3·67-s − 9·71-s − 5.85·73-s + 2.85·77-s + 14.9·79-s − 5.38·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.377·7-s − 0.860·11-s + 1.51·13-s + 0.877·17-s − 0.229·19-s − 0.466·23-s − 0.800·25-s + 1.13·29-s − 1.81·31-s − 0.169·35-s − 1.55·37-s + 0.0596·41-s − 0.304·43-s + 0.103·47-s + 0.142·49-s + 1.52·53-s − 0.384·55-s − 1.03·59-s − 0.640·61-s + 0.678·65-s − 1.87·67-s − 1.06·71-s − 0.685·73-s + 0.325·77-s + 1.68·79-s − 0.590·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 - T + 5T^{2} \) |
| 11 | \( 1 + 2.85T + 11T^{2} \) |
| 13 | \( 1 - 5.47T + 13T^{2} \) |
| 17 | \( 1 - 3.61T + 17T^{2} \) |
| 23 | \( 1 + 2.23T + 23T^{2} \) |
| 29 | \( 1 - 6.09T + 29T^{2} \) |
| 31 | \( 1 + 10.0T + 31T^{2} \) |
| 37 | \( 1 + 9.47T + 37T^{2} \) |
| 41 | \( 1 - 0.381T + 41T^{2} \) |
| 43 | \( 1 + 2T + 43T^{2} \) |
| 47 | \( 1 - 0.708T + 47T^{2} \) |
| 53 | \( 1 - 11.0T + 53T^{2} \) |
| 59 | \( 1 + 7.94T + 59T^{2} \) |
| 61 | \( 1 + 5T + 61T^{2} \) |
| 67 | \( 1 + 15.3T + 67T^{2} \) |
| 71 | \( 1 + 9T + 71T^{2} \) |
| 73 | \( 1 + 5.85T + 73T^{2} \) |
| 79 | \( 1 - 14.9T + 79T^{2} \) |
| 83 | \( 1 + 5.38T + 83T^{2} \) |
| 89 | \( 1 - 7.23T + 89T^{2} \) |
| 97 | \( 1 + 4.52T + 97T^{2} \) |
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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
−7.40152582777729722643150357514, −6.59017708368754238040335529643, −5.79682432142430521420368350166, −5.60309739469840712290416154037, −4.57341430749554621725338699665, −3.65269038143831425623329880508, −3.16613908814862365651645798412, −2.10263257003758193074506781278, −1.31519522073904260201629726352, 0,
1.31519522073904260201629726352, 2.10263257003758193074506781278, 3.16613908814862365651645798412, 3.65269038143831425623329880508, 4.57341430749554621725338699665, 5.60309739469840712290416154037, 5.79682432142430521420368350166, 6.59017708368754238040335529643, 7.40152582777729722643150357514
