| L(s) = 1 | − 3.70·5-s − 4.20·7-s + 1.09·11-s − 13-s − 0.298·17-s + 1.09·19-s + 8.70·25-s − 2·29-s + 5.13·31-s + 15.5·35-s + 3.70·37-s + 9.40·41-s − 5.29·43-s + 4.20·47-s + 10.7·49-s + 1.40·53-s − 4.04·55-s + 13.5·59-s − 9.40·61-s + 3.70·65-s − 11.3·67-s + 8.25·71-s − 6·73-s − 4.59·77-s + 14.6·79-s − 7.32·83-s + 1.10·85-s + ⋯ |
| L(s) = 1 | − 1.65·5-s − 1.59·7-s + 0.329·11-s − 0.277·13-s − 0.0723·17-s + 0.250·19-s + 1.74·25-s − 0.371·29-s + 0.922·31-s + 2.63·35-s + 0.608·37-s + 1.46·41-s − 0.808·43-s + 0.613·47-s + 1.52·49-s + 0.192·53-s − 0.545·55-s + 1.76·59-s − 1.20·61-s + 0.459·65-s − 1.38·67-s + 0.979·71-s − 0.702·73-s − 0.523·77-s + 1.64·79-s − 0.803·83-s + 0.119·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7488 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7488 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 + 3.70T + 5T^{2} \) |
| 7 | \( 1 + 4.20T + 7T^{2} \) |
| 11 | \( 1 - 1.09T + 11T^{2} \) |
| 17 | \( 1 + 0.298T + 17T^{2} \) |
| 19 | \( 1 - 1.09T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 - 5.13T + 31T^{2} \) |
| 37 | \( 1 - 3.70T + 37T^{2} \) |
| 41 | \( 1 - 9.40T + 41T^{2} \) |
| 43 | \( 1 + 5.29T + 43T^{2} \) |
| 47 | \( 1 - 4.20T + 47T^{2} \) |
| 53 | \( 1 - 1.40T + 53T^{2} \) |
| 59 | \( 1 - 13.5T + 59T^{2} \) |
| 61 | \( 1 + 9.40T + 61T^{2} \) |
| 67 | \( 1 + 11.3T + 67T^{2} \) |
| 71 | \( 1 - 8.25T + 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 - 14.6T + 79T^{2} \) |
| 83 | \( 1 + 7.32T + 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 - 8.80T + 97T^{2} \) |
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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
−7.56268788528399990108987945479, −6.86298008111048986441528573925, −6.36730065145126637756924858995, −5.46833489669855354840731042818, −4.41581608334191897435239196048, −3.92214676061834151179740557016, −3.22923874769557115535923425146, −2.57667567965592075528854474191, −0.903903083101652986774411356564, 0,
0.903903083101652986774411356564, 2.57667567965592075528854474191, 3.22923874769557115535923425146, 3.92214676061834151179740557016, 4.41581608334191897435239196048, 5.46833489669855354840731042818, 6.36730065145126637756924858995, 6.86298008111048986441528573925, 7.56268788528399990108987945479
