| L(s) = 1 | − 0.124·2-s − 1.98·4-s + 1.22·5-s + 2.66·7-s + 0.494·8-s − 0.152·10-s − 5.63·11-s + 0.0134·13-s − 0.331·14-s + 3.90·16-s + 3.54·17-s − 5.96·19-s − 2.43·20-s + 0.699·22-s − 4.08·23-s − 3.48·25-s − 0.00167·26-s − 5.29·28-s − 2.96·29-s − 1.47·32-s − 0.440·34-s + 3.27·35-s − 10.6·37-s + 0.740·38-s + 0.608·40-s + 7.14·41-s + 3.17·43-s + ⋯ |
| L(s) = 1 | − 0.0878·2-s − 0.992·4-s + 0.549·5-s + 1.00·7-s + 0.174·8-s − 0.0482·10-s − 1.69·11-s + 0.00373·13-s − 0.0885·14-s + 0.976·16-s + 0.859·17-s − 1.36·19-s − 0.545·20-s + 0.149·22-s − 0.852·23-s − 0.697·25-s − 0.000327·26-s − 1.00·28-s − 0.550·29-s − 0.260·32-s − 0.0754·34-s + 0.554·35-s − 1.74·37-s + 0.120·38-s + 0.0961·40-s + 1.11·41-s + 0.484·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8649 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8649 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.287349456\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.287349456\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 31 | \( 1 \) |
| good | 2 | \( 1 + 0.124T + 2T^{2} \) |
| 5 | \( 1 - 1.22T + 5T^{2} \) |
| 7 | \( 1 - 2.66T + 7T^{2} \) |
| 11 | \( 1 + 5.63T + 11T^{2} \) |
| 13 | \( 1 - 0.0134T + 13T^{2} \) |
| 17 | \( 1 - 3.54T + 17T^{2} \) |
| 19 | \( 1 + 5.96T + 19T^{2} \) |
| 23 | \( 1 + 4.08T + 23T^{2} \) |
| 29 | \( 1 + 2.96T + 29T^{2} \) |
| 37 | \( 1 + 10.6T + 37T^{2} \) |
| 41 | \( 1 - 7.14T + 41T^{2} \) |
| 43 | \( 1 - 3.17T + 43T^{2} \) |
| 47 | \( 1 - 9.81T + 47T^{2} \) |
| 53 | \( 1 - 7.90T + 53T^{2} \) |
| 59 | \( 1 - 10.3T + 59T^{2} \) |
| 61 | \( 1 + 3.70T + 61T^{2} \) |
| 67 | \( 1 - 3.56T + 67T^{2} \) |
| 71 | \( 1 + 3.76T + 71T^{2} \) |
| 73 | \( 1 - 3.55T + 73T^{2} \) |
| 79 | \( 1 - 3.25T + 79T^{2} \) |
| 83 | \( 1 - 2.07T + 83T^{2} \) |
| 89 | \( 1 - 8.80T + 89T^{2} \) |
| 97 | \( 1 - 3.38T + 97T^{2} \) |
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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
−7.77090902843174910414049215851, −7.46824469319704634267542667828, −6.14400910415514920642255325088, −5.48993305484406576498924994240, −5.15665814983963622259568381622, −4.30912006638192856543578154613, −3.65212862561397292735978646980, −2.43134954104593420521596478485, −1.83589537247644082801252877943, −0.55482461347202711219580912465,
0.55482461347202711219580912465, 1.83589537247644082801252877943, 2.43134954104593420521596478485, 3.65212862561397292735978646980, 4.30912006638192856543578154613, 5.15665814983963622259568381622, 5.48993305484406576498924994240, 6.14400910415514920642255325088, 7.46824469319704634267542667828, 7.77090902843174910414049215851
