| L(s) = 1 | + 0.267·5-s + 7-s − 6.19·11-s − 6.46·13-s + 7·17-s − 0.732·19-s + 4.19·23-s − 4.92·25-s + 1.53·29-s − 8.19·31-s + 0.267·35-s + 10.6·37-s + 2.53·41-s + 1.46·43-s − 4.73·47-s + 49-s − 9.46·53-s − 1.66·55-s − 4.19·59-s + 3.92·61-s − 1.73·65-s + 6.73·67-s − 6.53·71-s + 8.26·73-s − 6.19·77-s + 9.12·79-s − 16.5·83-s + ⋯ |
| L(s) = 1 | + 0.119·5-s + 0.377·7-s − 1.86·11-s − 1.79·13-s + 1.69·17-s − 0.167·19-s + 0.874·23-s − 0.985·25-s + 0.285·29-s − 1.47·31-s + 0.0452·35-s + 1.75·37-s + 0.396·41-s + 0.223·43-s − 0.690·47-s + 0.142·49-s − 1.29·53-s − 0.223·55-s − 0.546·59-s + 0.502·61-s − 0.214·65-s + 0.822·67-s − 0.775·71-s + 0.967·73-s − 0.706·77-s + 1.02·79-s − 1.82·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9072 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.405363807\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.405363807\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 5 | \( 1 - 0.267T + 5T^{2} \) |
| 11 | \( 1 + 6.19T + 11T^{2} \) |
| 13 | \( 1 + 6.46T + 13T^{2} \) |
| 17 | \( 1 - 7T + 17T^{2} \) |
| 19 | \( 1 + 0.732T + 19T^{2} \) |
| 23 | \( 1 - 4.19T + 23T^{2} \) |
| 29 | \( 1 - 1.53T + 29T^{2} \) |
| 31 | \( 1 + 8.19T + 31T^{2} \) |
| 37 | \( 1 - 10.6T + 37T^{2} \) |
| 41 | \( 1 - 2.53T + 41T^{2} \) |
| 43 | \( 1 - 1.46T + 43T^{2} \) |
| 47 | \( 1 + 4.73T + 47T^{2} \) |
| 53 | \( 1 + 9.46T + 53T^{2} \) |
| 59 | \( 1 + 4.19T + 59T^{2} \) |
| 61 | \( 1 - 3.92T + 61T^{2} \) |
| 67 | \( 1 - 6.73T + 67T^{2} \) |
| 71 | \( 1 + 6.53T + 71T^{2} \) |
| 73 | \( 1 - 8.26T + 73T^{2} \) |
| 79 | \( 1 - 9.12T + 79T^{2} \) |
| 83 | \( 1 + 16.5T + 83T^{2} \) |
| 89 | \( 1 - 9.92T + 89T^{2} \) |
| 97 | \( 1 - 10.9T + 97T^{2} \) |
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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
−7.60372964073524971568189376367, −7.48128645895252465497356083019, −6.29008160151389394739045009157, −5.34609259631136655877147652103, −5.21979017060953460268006433952, −4.39748627905835612663319986734, −3.25195937365871008454637067838, −2.63702114579664216814729533196, −1.89019012697180472648808454586, −0.54774047136431459847219391772,
0.54774047136431459847219391772, 1.89019012697180472648808454586, 2.63702114579664216814729533196, 3.25195937365871008454637067838, 4.39748627905835612663319986734, 5.21979017060953460268006433952, 5.34609259631136655877147652103, 6.29008160151389394739045009157, 7.48128645895252465497356083019, 7.60372964073524971568189376367
