| L(s) = 1 | + 5-s − 4.67·7-s + 0.672·11-s − 1.14·13-s − 3.52·17-s + 5.52·19-s + 3.81·23-s + 25-s + 29-s − 1.52·31-s − 4.67·35-s + 7.16·37-s − 2.85·41-s + 8.96·43-s + 6.67·47-s + 14.8·49-s − 10.4·53-s + 0.672·55-s − 10.7·59-s − 14.4·61-s − 1.14·65-s − 7.81·67-s − 4.48·71-s − 4.96·73-s − 3.14·77-s + 2.38·79-s − 14.0·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.76·7-s + 0.202·11-s − 0.317·13-s − 0.855·17-s + 1.26·19-s + 0.795·23-s + 0.200·25-s + 0.185·29-s − 0.274·31-s − 0.789·35-s + 1.17·37-s − 0.446·41-s + 1.36·43-s + 0.973·47-s + 2.11·49-s − 1.44·53-s + 0.0907·55-s − 1.40·59-s − 1.85·61-s − 0.141·65-s − 0.954·67-s − 0.532·71-s − 0.580·73-s − 0.358·77-s + 0.268·79-s − 1.53·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5220 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 7 | \( 1 + 4.67T + 7T^{2} \) |
| 11 | \( 1 - 0.672T + 11T^{2} \) |
| 13 | \( 1 + 1.14T + 13T^{2} \) |
| 17 | \( 1 + 3.52T + 17T^{2} \) |
| 19 | \( 1 - 5.52T + 19T^{2} \) |
| 23 | \( 1 - 3.81T + 23T^{2} \) |
| 31 | \( 1 + 1.52T + 31T^{2} \) |
| 37 | \( 1 - 7.16T + 37T^{2} \) |
| 41 | \( 1 + 2.85T + 41T^{2} \) |
| 43 | \( 1 - 8.96T + 43T^{2} \) |
| 47 | \( 1 - 6.67T + 47T^{2} \) |
| 53 | \( 1 + 10.4T + 53T^{2} \) |
| 59 | \( 1 + 10.7T + 59T^{2} \) |
| 61 | \( 1 + 14.4T + 61T^{2} \) |
| 67 | \( 1 + 7.81T + 67T^{2} \) |
| 71 | \( 1 + 4.48T + 71T^{2} \) |
| 73 | \( 1 + 4.96T + 73T^{2} \) |
| 79 | \( 1 - 2.38T + 79T^{2} \) |
| 83 | \( 1 + 14.0T + 83T^{2} \) |
| 89 | \( 1 - 1.63T + 89T^{2} \) |
| 97 | \( 1 + 9.32T + 97T^{2} \) |
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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
−7.59856827358761333804860977260, −7.14200907682236930242847764227, −6.26439082449963576541415048603, −5.95375970594210402773554882291, −4.92846684637236090465011592352, −4.07560427281879546221822867948, −3.05966648970538958709406427185, −2.68416228837162531807896907641, −1.27692092150067006038620236935, 0,
1.27692092150067006038620236935, 2.68416228837162531807896907641, 3.05966648970538958709406427185, 4.07560427281879546221822867948, 4.92846684637236090465011592352, 5.95375970594210402773554882291, 6.26439082449963576541415048603, 7.14200907682236930242847764227, 7.59856827358761333804860977260
