| L(s) = 1 | − 16.6·5-s + 71.7·11-s + 65.3·13-s + 90.4·17-s − 163.·19-s − 79.2·23-s + 152.·25-s + 43.2·29-s − 135.·31-s + 270.·37-s + 152.·41-s − 177.·43-s − 45.6·47-s + 158.·53-s − 1.19e3·55-s − 391.·59-s − 551.·61-s − 1.08e3·65-s + 458.·67-s + 486.·71-s − 574.·73-s − 668.·79-s + 76.2·83-s − 1.50e3·85-s + 1.36e3·89-s + 2.72e3·95-s − 242.·97-s + ⋯ |
| L(s) = 1 | − 1.48·5-s + 1.96·11-s + 1.39·13-s + 1.29·17-s − 1.97·19-s − 0.718·23-s + 1.21·25-s + 0.277·29-s − 0.785·31-s + 1.20·37-s + 0.579·41-s − 0.630·43-s − 0.141·47-s + 0.410·53-s − 2.93·55-s − 0.864·59-s − 1.15·61-s − 2.07·65-s + 0.836·67-s + 0.813·71-s − 0.921·73-s − 0.951·79-s + 0.100·83-s − 1.92·85-s + 1.62·89-s + 2.94·95-s − 0.253·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.791645869\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.791645869\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 16.6T + 125T^{2} \) |
| 11 | \( 1 - 71.7T + 1.33e3T^{2} \) |
| 13 | \( 1 - 65.3T + 2.19e3T^{2} \) |
| 17 | \( 1 - 90.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 163.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 79.2T + 1.21e4T^{2} \) |
| 29 | \( 1 - 43.2T + 2.43e4T^{2} \) |
| 31 | \( 1 + 135.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 270.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 152.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 177.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 45.6T + 1.03e5T^{2} \) |
| 53 | \( 1 - 158.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 391.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 551.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 458.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 486.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 574.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 668.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 76.2T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.36e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 242.T + 9.12e5T^{2} \) |
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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
−8.722327132069747561656774154588, −8.254564342853944540200622828424, −7.44311914849651945119178867674, −6.47163007628696395359022572790, −5.98709077833065777667266468589, −4.40929974488391429139251651484, −3.95400239308725660352877293616, −3.33795175033010433243333321443, −1.66128963728593944314225337605, −0.66788402554379578868326489412,
0.66788402554379578868326489412, 1.66128963728593944314225337605, 3.33795175033010433243333321443, 3.95400239308725660352877293616, 4.40929974488391429139251651484, 5.98709077833065777667266468589, 6.47163007628696395359022572790, 7.44311914849651945119178867674, 8.254564342853944540200622828424, 8.722327132069747561656774154588
