| L(s) = 1 | − 2.49·5-s + 0.917·7-s − 3.69·11-s + 5.81·13-s + 0.867·17-s + 6.52·19-s − 4·23-s + 1.23·25-s + 7.72·29-s − 2.14·31-s − 2.29·35-s − 2.47·37-s + 9.58·41-s − 9.58·43-s + 1.65·47-s − 6.15·49-s − 3.39·53-s + 9.22·55-s + 12.7·59-s + 0.0231·61-s − 14.5·65-s + 5.32·67-s − 11.8·71-s − 15.2·73-s − 3.39·77-s + 8.40·79-s − 1.96·83-s + ⋯ |
| L(s) = 1 | − 1.11·5-s + 0.346·7-s − 1.11·11-s + 1.61·13-s + 0.210·17-s + 1.49·19-s − 0.834·23-s + 0.246·25-s + 1.43·29-s − 0.385·31-s − 0.387·35-s − 0.406·37-s + 1.49·41-s − 1.46·43-s + 0.241·47-s − 0.879·49-s − 0.466·53-s + 1.24·55-s + 1.65·59-s + 0.00296·61-s − 1.79·65-s + 0.650·67-s − 1.40·71-s − 1.78·73-s − 0.386·77-s + 0.946·79-s − 0.215·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.585579118\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.585579118\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.49T + 5T^{2} \) |
| 7 | \( 1 - 0.917T + 7T^{2} \) |
| 11 | \( 1 + 3.69T + 11T^{2} \) |
| 13 | \( 1 - 5.81T + 13T^{2} \) |
| 17 | \( 1 - 0.867T + 17T^{2} \) |
| 19 | \( 1 - 6.52T + 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 - 7.72T + 29T^{2} \) |
| 31 | \( 1 + 2.14T + 31T^{2} \) |
| 37 | \( 1 + 2.47T + 37T^{2} \) |
| 41 | \( 1 - 9.58T + 41T^{2} \) |
| 43 | \( 1 + 9.58T + 43T^{2} \) |
| 47 | \( 1 - 1.65T + 47T^{2} \) |
| 53 | \( 1 + 3.39T + 53T^{2} \) |
| 59 | \( 1 - 12.7T + 59T^{2} \) |
| 61 | \( 1 - 0.0231T + 61T^{2} \) |
| 67 | \( 1 - 5.32T + 67T^{2} \) |
| 71 | \( 1 + 11.8T + 71T^{2} \) |
| 73 | \( 1 + 15.2T + 73T^{2} \) |
| 79 | \( 1 - 8.40T + 79T^{2} \) |
| 83 | \( 1 + 1.96T + 83T^{2} \) |
| 89 | \( 1 + 2.79T + 89T^{2} \) |
| 97 | \( 1 + 2.26T + 97T^{2} \) |
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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
−7.76386610613147697966380325341, −7.26856750538946539530627695835, −6.32269513227229175077072632366, −5.62570114987916399256750171418, −4.93818542188093410228966966691, −4.14092675074401816023401618332, −3.46666469377130962932680451951, −2.84249248328985699291233684837, −1.59596133706480409364298775101, −0.62900031308888376797520129706,
0.62900031308888376797520129706, 1.59596133706480409364298775101, 2.84249248328985699291233684837, 3.46666469377130962932680451951, 4.14092675074401816023401618332, 4.93818542188093410228966966691, 5.62570114987916399256750171418, 6.32269513227229175077072632366, 7.26856750538946539530627695835, 7.76386610613147697966380325341
