| L(s) = 1 | + 2.64·5-s + 0.913·7-s + 5.58·11-s − 2.58·13-s + 1.73·17-s − 7.84·19-s + 1.58·23-s + 2.00·25-s − 0.818·29-s + 1.82·31-s + 2.41·35-s + 6.58·37-s + 8.75·41-s + 7.84·43-s − 8·47-s − 6.16·49-s + 8.75·53-s + 14.7·55-s + 8·59-s + 10.5·61-s − 6.83·65-s + 7.84·67-s − 9.58·71-s + 12.1·73-s + 5.10·77-s − 14.7·79-s − 3.16·83-s + ⋯ |
| L(s) = 1 | + 1.18·5-s + 0.345·7-s + 1.68·11-s − 0.716·13-s + 0.420·17-s − 1.79·19-s + 0.329·23-s + 0.400·25-s − 0.151·29-s + 0.328·31-s + 0.408·35-s + 1.08·37-s + 1.36·41-s + 1.19·43-s − 1.16·47-s − 0.880·49-s + 1.20·53-s + 1.99·55-s + 1.04·59-s + 1.35·61-s − 0.847·65-s + 0.958·67-s − 1.13·71-s + 1.42·73-s + 0.581·77-s − 1.66·79-s − 0.347·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.891088069\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.891088069\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 2.64T + 5T^{2} \) |
| 7 | \( 1 - 0.913T + 7T^{2} \) |
| 11 | \( 1 - 5.58T + 11T^{2} \) |
| 13 | \( 1 + 2.58T + 13T^{2} \) |
| 17 | \( 1 - 1.73T + 17T^{2} \) |
| 19 | \( 1 + 7.84T + 19T^{2} \) |
| 23 | \( 1 - 1.58T + 23T^{2} \) |
| 29 | \( 1 + 0.818T + 29T^{2} \) |
| 31 | \( 1 - 1.82T + 31T^{2} \) |
| 37 | \( 1 - 6.58T + 37T^{2} \) |
| 41 | \( 1 - 8.75T + 41T^{2} \) |
| 43 | \( 1 - 7.84T + 43T^{2} \) |
| 47 | \( 1 + 8T + 47T^{2} \) |
| 53 | \( 1 - 8.75T + 53T^{2} \) |
| 59 | \( 1 - 8T + 59T^{2} \) |
| 61 | \( 1 - 10.5T + 61T^{2} \) |
| 67 | \( 1 - 7.84T + 67T^{2} \) |
| 71 | \( 1 + 9.58T + 71T^{2} \) |
| 73 | \( 1 - 12.1T + 73T^{2} \) |
| 79 | \( 1 + 14.7T + 79T^{2} \) |
| 83 | \( 1 + 3.16T + 83T^{2} \) |
| 89 | \( 1 - 8.85T + 89T^{2} \) |
| 97 | \( 1 - 13.1T + 97T^{2} \) |
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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
−8.368108729419357538359258679806, −7.40225529863397961933967385987, −6.58869643572600506939213107479, −6.16117388991053449488901382740, −5.40729889303814265814681597149, −4.46187795318553184188180150192, −3.89089179871664383853826823324, −2.58108536845864651344886536714, −1.94257672624049055142084632179, −0.970517762708426154486035032372,
0.970517762708426154486035032372, 1.94257672624049055142084632179, 2.58108536845864651344886536714, 3.89089179871664383853826823324, 4.46187795318553184188180150192, 5.40729889303814265814681597149, 6.16117388991053449488901382740, 6.58869643572600506939213107479, 7.40225529863397961933967385987, 8.368108729419357538359258679806
