| L(s) = 1 | − 2-s + 4-s + 2.89·5-s + 7-s − 8-s − 2.89·10-s − 11-s − 0.364·13-s − 14-s + 16-s + 2.89·17-s + 7.25·19-s + 2.89·20-s + 22-s − 8.14·23-s + 3.36·25-s + 0.364·26-s + 28-s + 2.36·29-s + 10.6·31-s − 32-s − 2.89·34-s + 2.89·35-s − 3.78·37-s − 7.25·38-s − 2.89·40-s − 2.89·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 1.29·5-s + 0.377·7-s − 0.353·8-s − 0.914·10-s − 0.301·11-s − 0.101·13-s − 0.267·14-s + 0.250·16-s + 0.701·17-s + 1.66·19-s + 0.646·20-s + 0.213·22-s − 1.69·23-s + 0.672·25-s + 0.0715·26-s + 0.188·28-s + 0.439·29-s + 1.91·31-s − 0.176·32-s − 0.496·34-s + 0.488·35-s − 0.622·37-s − 1.17·38-s − 0.457·40-s − 0.451·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.660501802\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.660501802\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| good | 5 | \( 1 - 2.89T + 5T^{2} \) |
| 13 | \( 1 + 0.364T + 13T^{2} \) |
| 17 | \( 1 - 2.89T + 17T^{2} \) |
| 19 | \( 1 - 7.25T + 19T^{2} \) |
| 23 | \( 1 + 8.14T + 23T^{2} \) |
| 29 | \( 1 - 2.36T + 29T^{2} \) |
| 31 | \( 1 - 10.6T + 31T^{2} \) |
| 37 | \( 1 + 3.78T + 37T^{2} \) |
| 41 | \( 1 + 2.89T + 41T^{2} \) |
| 43 | \( 1 - 11.4T + 43T^{2} \) |
| 47 | \( 1 - 6.52T + 47T^{2} \) |
| 53 | \( 1 + 11.7T + 53T^{2} \) |
| 59 | \( 1 - 10.5T + 59T^{2} \) |
| 61 | \( 1 + 11.9T + 61T^{2} \) |
| 67 | \( 1 + 6.14T + 67T^{2} \) |
| 71 | \( 1 + 13.9T + 71T^{2} \) |
| 73 | \( 1 + 1.10T + 73T^{2} \) |
| 79 | \( 1 - 10.1T + 79T^{2} \) |
| 83 | \( 1 - 3.10T + 83T^{2} \) |
| 89 | \( 1 + 2.14T + 89T^{2} \) |
| 97 | \( 1 - 6.72T + 97T^{2} \) |
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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
−9.848092349667463015734341237262, −8.872128165583520140954299731368, −7.968853825868422455870902964811, −7.34877622062011195067394977911, −6.14256330148816180868122592495, −5.70375203652779360508893419870, −4.65813304757256387640730117019, −3.14513020988049816686614276049, −2.16132774926133614899209265429, −1.11638262699769529792801571224,
1.11638262699769529792801571224, 2.16132774926133614899209265429, 3.14513020988049816686614276049, 4.65813304757256387640730117019, 5.70375203652779360508893419870, 6.14256330148816180868122592495, 7.34877622062011195067394977911, 7.968853825868422455870902964811, 8.872128165583520140954299731368, 9.848092349667463015734341237262
