| L(s) = 1 | + 4.75·2-s + 3·3-s + 14.6·4-s − 7.65·5-s + 14.2·6-s − 31.5·7-s + 31.6·8-s + 9·9-s − 36.4·10-s − 7.18·11-s + 43.9·12-s + 84.3·13-s − 150.·14-s − 22.9·15-s + 33.5·16-s + 17·17-s + 42.8·18-s − 37.0·19-s − 112.·20-s − 94.7·21-s − 34.2·22-s + 150.·23-s + 95.0·24-s − 66.3·25-s + 401.·26-s + 27·27-s − 462.·28-s + ⋯ |
| L(s) = 1 | + 1.68·2-s + 0.577·3-s + 1.83·4-s − 0.684·5-s + 0.971·6-s − 1.70·7-s + 1.40·8-s + 0.333·9-s − 1.15·10-s − 0.197·11-s + 1.05·12-s + 1.79·13-s − 2.87·14-s − 0.395·15-s + 0.524·16-s + 0.242·17-s + 0.560·18-s − 0.447·19-s − 1.25·20-s − 0.984·21-s − 0.331·22-s + 1.36·23-s + 0.808·24-s − 0.531·25-s + 3.02·26-s + 0.192·27-s − 3.12·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.080549932\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.080549932\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 17 | \( 1 - 17T \) |
| good | 2 | \( 1 - 4.75T + 8T^{2} \) |
| 5 | \( 1 + 7.65T + 125T^{2} \) |
| 7 | \( 1 + 31.5T + 343T^{2} \) |
| 11 | \( 1 + 7.18T + 1.33e3T^{2} \) |
| 13 | \( 1 - 84.3T + 2.19e3T^{2} \) |
| 19 | \( 1 + 37.0T + 6.85e3T^{2} \) |
| 23 | \( 1 - 150.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 11.5T + 2.43e4T^{2} \) |
| 31 | \( 1 + 53.2T + 2.97e4T^{2} \) |
| 37 | \( 1 + 99.2T + 5.06e4T^{2} \) |
| 41 | \( 1 - 118.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 456.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 571.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 462.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 48.0T + 2.05e5T^{2} \) |
| 61 | \( 1 - 59.5T + 2.26e5T^{2} \) |
| 67 | \( 1 + 740.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 930.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 697.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.03e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 22.2T + 5.71e5T^{2} \) |
| 89 | \( 1 + 369.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.13e3T + 9.12e5T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.00534743570540833556941317998, −13.48337910026184903997176019376, −13.15831831979228801890627319657, −12.01112842167016326974114611538, −10.65569400918044152494048314730, −8.891722224660689997487696992764, −7.05110697961841936994482415663, −5.92696068563719452301435510512, −3.93993670382707110859565914954, −3.13224268386421413321316067651,
3.13224268386421413321316067651, 3.93993670382707110859565914954, 5.92696068563719452301435510512, 7.05110697961841936994482415663, 8.891722224660689997487696992764, 10.65569400918044152494048314730, 12.01112842167016326974114611538, 13.15831831979228801890627319657, 13.48337910026184903997176019376, 15.00534743570540833556941317998
