| L(s) = 1 | − 2.15·5-s − 3.37·7-s − 4.70·11-s + 2·13-s − 2.15·17-s − 19-s − 6.85·23-s − 0.372·25-s + 6.85·29-s + 6.74·31-s + 7.25·35-s − 0.744·37-s + 2.55·41-s − 6.11·43-s + 9.00·47-s + 4.37·49-s − 11.9·53-s + 10.1·55-s + 5.10·59-s + 12.1·61-s − 4.30·65-s + 4·67-s − 13.7·71-s + 12.1·73-s + 15.8·77-s + 4·79-s + 1.75·83-s + ⋯ |
| L(s) = 1 | − 0.962·5-s − 1.27·7-s − 1.41·11-s + 0.554·13-s − 0.521·17-s − 0.229·19-s − 1.42·23-s − 0.0744·25-s + 1.27·29-s + 1.21·31-s + 1.22·35-s − 0.122·37-s + 0.398·41-s − 0.932·43-s + 1.31·47-s + 0.624·49-s − 1.64·53-s + 1.36·55-s + 0.664·59-s + 1.55·61-s − 0.533·65-s + 0.488·67-s − 1.62·71-s + 1.41·73-s + 1.80·77-s + 0.450·79-s + 0.192·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6718268525\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6718268525\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 + 2.15T + 5T^{2} \) |
| 7 | \( 1 + 3.37T + 7T^{2} \) |
| 11 | \( 1 + 4.70T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + 2.15T + 17T^{2} \) |
| 23 | \( 1 + 6.85T + 23T^{2} \) |
| 29 | \( 1 - 6.85T + 29T^{2} \) |
| 31 | \( 1 - 6.74T + 31T^{2} \) |
| 37 | \( 1 + 0.744T + 37T^{2} \) |
| 41 | \( 1 - 2.55T + 41T^{2} \) |
| 43 | \( 1 + 6.11T + 43T^{2} \) |
| 47 | \( 1 - 9.00T + 47T^{2} \) |
| 53 | \( 1 + 11.9T + 53T^{2} \) |
| 59 | \( 1 - 5.10T + 59T^{2} \) |
| 61 | \( 1 - 12.1T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 + 13.7T + 71T^{2} \) |
| 73 | \( 1 - 12.1T + 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 - 1.75T + 83T^{2} \) |
| 89 | \( 1 + 11.9T + 89T^{2} \) |
| 97 | \( 1 + 15.4T + 97T^{2} \) |
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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
−8.542703413663872604287355756967, −8.188085705404294572356566101304, −7.36396442346700714759525199173, −6.49794453748563139172271638055, −5.91481280872669157378273828487, −4.79611241091220360371901940495, −3.97823690290518255640872029998, −3.17966453059115101880052687183, −2.33466469563871227316883924625, −0.47879729343034893960514321785,
0.47879729343034893960514321785, 2.33466469563871227316883924625, 3.17966453059115101880052687183, 3.97823690290518255640872029998, 4.79611241091220360371901940495, 5.91481280872669157378273828487, 6.49794453748563139172271638055, 7.36396442346700714759525199173, 8.188085705404294572356566101304, 8.542703413663872604287355756967
