| L(s) = 1 | − 1.31·5-s − 3.04·7-s − 1.31·13-s + 6.27·17-s + 3.04·19-s − 3.27·25-s − 6.27·29-s + 5.27·31-s + 4·35-s + 5.54·37-s − 2.27·41-s + 0.837·43-s − 12.1·47-s + 2.27·49-s − 5.61·53-s − 5.25·59-s − 3.04·61-s + 1.72·65-s − 7.82·67-s + 6.92·71-s + 3.88·73-s − 8.29·79-s + 12·83-s − 8.24·85-s + 1.31·89-s + 4·91-s − 4·95-s + ⋯ |
| L(s) = 1 | − 0.587·5-s − 1.15·7-s − 0.364·13-s + 1.52·17-s + 0.698·19-s − 0.654·25-s − 1.16·29-s + 0.947·31-s + 0.676·35-s + 0.912·37-s − 0.355·41-s + 0.127·43-s − 1.77·47-s + 0.324·49-s − 0.771·53-s − 0.683·59-s − 0.389·61-s + 0.213·65-s − 0.955·67-s + 0.822·71-s + 0.454·73-s − 0.933·79-s + 1.31·83-s − 0.893·85-s + 0.139·89-s + 0.419·91-s − 0.410·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.135480851\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.135480851\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 + 1.31T + 5T^{2} \) |
| 7 | \( 1 + 3.04T + 7T^{2} \) |
| 13 | \( 1 + 1.31T + 13T^{2} \) |
| 17 | \( 1 - 6.27T + 17T^{2} \) |
| 19 | \( 1 - 3.04T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 6.27T + 29T^{2} \) |
| 31 | \( 1 - 5.27T + 31T^{2} \) |
| 37 | \( 1 - 5.54T + 37T^{2} \) |
| 41 | \( 1 + 2.27T + 41T^{2} \) |
| 43 | \( 1 - 0.837T + 43T^{2} \) |
| 47 | \( 1 + 12.1T + 47T^{2} \) |
| 53 | \( 1 + 5.61T + 53T^{2} \) |
| 59 | \( 1 + 5.25T + 59T^{2} \) |
| 61 | \( 1 + 3.04T + 61T^{2} \) |
| 67 | \( 1 + 7.82T + 67T^{2} \) |
| 71 | \( 1 - 6.92T + 71T^{2} \) |
| 73 | \( 1 - 3.88T + 73T^{2} \) |
| 79 | \( 1 + 8.29T + 79T^{2} \) |
| 83 | \( 1 - 12T + 83T^{2} \) |
| 89 | \( 1 - 1.31T + 89T^{2} \) |
| 97 | \( 1 + 15.5T + 97T^{2} \) |
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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
−7.80606050217096592130069540070, −7.15775348924028780577488287346, −6.36890560264457089088892639665, −5.77998517897829369417085157699, −5.01468728746535713548729834339, −4.13114680186515266850726202369, −3.31074237300219237251073396586, −2.99272959854778258878002629742, −1.68400091852666222331358242753, −0.51566352257393830610751226795,
0.51566352257393830610751226795, 1.68400091852666222331358242753, 2.99272959854778258878002629742, 3.31074237300219237251073396586, 4.13114680186515266850726202369, 5.01468728746535713548729834339, 5.77998517897829369417085157699, 6.36890560264457089088892639665, 7.15775348924028780577488287346, 7.80606050217096592130069540070
