| L(s) = 1 | + 2.82·5-s − 2.82·7-s − 2·11-s + 13-s + 3.65·17-s − 2.82·19-s + 4·23-s + 3.00·25-s + 2·29-s − 6.82·31-s − 8.00·35-s − 3.65·37-s − 10.8·41-s − 9.65·43-s + 0.343·47-s + 1.00·49-s − 2·53-s − 5.65·55-s − 3.65·59-s + 9.31·61-s + 2.82·65-s − 1.17·67-s − 2·71-s + 11.6·73-s + 5.65·77-s + 11.3·79-s − 7.65·83-s + ⋯ |
| L(s) = 1 | + 1.26·5-s − 1.06·7-s − 0.603·11-s + 0.277·13-s + 0.886·17-s − 0.648·19-s + 0.834·23-s + 0.600·25-s + 0.371·29-s − 1.22·31-s − 1.35·35-s − 0.601·37-s − 1.69·41-s − 1.47·43-s + 0.0500·47-s + 0.142·49-s − 0.274·53-s − 0.762·55-s − 0.476·59-s + 1.19·61-s + 0.350·65-s − 0.143·67-s − 0.237·71-s + 1.36·73-s + 0.644·77-s + 1.27·79-s − 0.840·83-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 | \( 1 - 2.82T + 5T^{2} \) |
| 7 | \( 1 + 2.82T + 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 17 | \( 1 - 3.65T + 17T^{2} \) |
| 19 | \( 1 + 2.82T + 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + 6.82T + 31T^{2} \) |
| 37 | \( 1 + 3.65T + 37T^{2} \) |
| 41 | \( 1 + 10.8T + 41T^{2} \) |
| 43 | \( 1 + 9.65T + 43T^{2} \) |
| 47 | \( 1 - 0.343T + 47T^{2} \) |
| 53 | \( 1 + 2T + 53T^{2} \) |
| 59 | \( 1 + 3.65T + 59T^{2} \) |
| 61 | \( 1 - 9.31T + 61T^{2} \) |
| 67 | \( 1 + 1.17T + 67T^{2} \) |
| 71 | \( 1 + 2T + 71T^{2} \) |
| 73 | \( 1 - 11.6T + 73T^{2} \) |
| 79 | \( 1 - 11.3T + 79T^{2} \) |
| 83 | \( 1 + 7.65T + 83T^{2} \) |
| 89 | \( 1 + 9.17T + 89T^{2} \) |
| 97 | \( 1 + 7.65T + 97T^{2} \) |
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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
−7.42288805120005157617126359742, −6.63568150208593029229226485823, −6.27352120781029551437112790242, −5.36716580915912102267263174021, −5.05836116475561885098897298631, −3.69289474811628930494470789311, −3.13950386261738642154802044268, −2.24774636445659575553930432772, −1.40056564454665554499467224673, 0,
1.40056564454665554499467224673, 2.24774636445659575553930432772, 3.13950386261738642154802044268, 3.69289474811628930494470789311, 5.05836116475561885098897298631, 5.36716580915912102267263174021, 6.27352120781029551437112790242, 6.63568150208593029229226485823, 7.42288805120005157617126359742
