| L(s) = 1 | − 2-s + 4-s + 1.64·5-s + 2·7-s − 8-s − 1.64·10-s − 1.64·11-s + 5.29·13-s − 2·14-s + 16-s − 3.29·17-s + 0.354·19-s + 1.64·20-s + 1.64·22-s + 23-s − 2.29·25-s − 5.29·26-s + 2·28-s + 9.29·29-s − 1.29·31-s − 32-s + 3.29·34-s + 3.29·35-s + 6.93·37-s − 0.354·38-s − 1.64·40-s + 6·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 0.736·5-s + 0.755·7-s − 0.353·8-s − 0.520·10-s − 0.496·11-s + 1.46·13-s − 0.534·14-s + 0.250·16-s − 0.798·17-s + 0.0812·19-s + 0.368·20-s + 0.350·22-s + 0.208·23-s − 0.458·25-s − 1.03·26-s + 0.377·28-s + 1.72·29-s − 0.231·31-s − 0.176·32-s + 0.564·34-s + 0.556·35-s + 1.14·37-s − 0.0574·38-s − 0.260·40-s + 0.937·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.245354469\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.245354469\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 - 1.64T + 5T^{2} \) |
| 7 | \( 1 - 2T + 7T^{2} \) |
| 11 | \( 1 + 1.64T + 11T^{2} \) |
| 13 | \( 1 - 5.29T + 13T^{2} \) |
| 17 | \( 1 + 3.29T + 17T^{2} \) |
| 19 | \( 1 - 0.354T + 19T^{2} \) |
| 29 | \( 1 - 9.29T + 29T^{2} \) |
| 31 | \( 1 + 1.29T + 31T^{2} \) |
| 37 | \( 1 - 6.93T + 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 - 0.354T + 43T^{2} \) |
| 47 | \( 1 - 6T + 47T^{2} \) |
| 53 | \( 1 + 1.64T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 0.354T + 61T^{2} \) |
| 67 | \( 1 + 14.9T + 67T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 + 7.29T + 73T^{2} \) |
| 79 | \( 1 - 8.58T + 79T^{2} \) |
| 83 | \( 1 + 13.6T + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 1.29T + 97T^{2} \) |
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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
−10.98155484782329337888949113709, −10.40482599114974020735257880798, −9.312501618093317564853641216025, −8.542776875790259506043258999385, −7.74513055401376614517540185136, −6.48938586343200272739556563697, −5.69890845920604040060471151389, −4.36398200018279965241270343877, −2.67311810030499191184477268671, −1.36365755960770692493242196130,
1.36365755960770692493242196130, 2.67311810030499191184477268671, 4.36398200018279965241270343877, 5.69890845920604040060471151389, 6.48938586343200272739556563697, 7.74513055401376614517540185136, 8.542776875790259506043258999385, 9.312501618093317564853641216025, 10.40482599114974020735257880798, 10.98155484782329337888949113709
