| L(s) = 1 | − 2·2-s + 10.0·3-s + 4·4-s − 20.1·6-s + 0.670·7-s − 8·8-s + 74.0·9-s − 15.7·11-s + 40.2·12-s + 50.2·13-s − 1.34·14-s + 16·16-s + 43.0·17-s − 148.·18-s + 86.7·19-s + 6.73·21-s + 31.4·22-s − 23·23-s − 80.4·24-s − 100.·26-s + 473.·27-s + 2.68·28-s + 147.·29-s − 121.·31-s − 32·32-s − 157.·33-s − 86.1·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.93·3-s + 0.5·4-s − 1.36·6-s + 0.0361·7-s − 0.353·8-s + 2.74·9-s − 0.430·11-s + 0.967·12-s + 1.07·13-s − 0.0255·14-s + 0.250·16-s + 0.614·17-s − 1.93·18-s + 1.04·19-s + 0.0700·21-s + 0.304·22-s − 0.208·23-s − 0.684·24-s − 0.758·26-s + 3.37·27-s + 0.0180·28-s + 0.941·29-s − 0.702·31-s − 0.176·32-s − 0.833·33-s − 0.434·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.754611437\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.754611437\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 5 | \( 1 \) |
| 23 | \( 1 + 23T \) |
| good | 3 | \( 1 - 10.0T + 27T^{2} \) |
| 7 | \( 1 - 0.670T + 343T^{2} \) |
| 11 | \( 1 + 15.7T + 1.33e3T^{2} \) |
| 13 | \( 1 - 50.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 43.0T + 4.91e3T^{2} \) |
| 19 | \( 1 - 86.7T + 6.85e3T^{2} \) |
| 29 | \( 1 - 147.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 121.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 241.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 232.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 302.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 252.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 58.0T + 1.48e5T^{2} \) |
| 59 | \( 1 + 375.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 353.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 750.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 29.4T + 3.57e5T^{2} \) |
| 73 | \( 1 - 80.5T + 3.89e5T^{2} \) |
| 79 | \( 1 + 607.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.23e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 903.T + 7.04e5T^{2} \) |
| 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
−9.292609418732479150018436473094, −8.550755747648764237970653553064, −8.011150810699143959153205853568, −7.37377737977104503172954326955, −6.43580847295372532944340458860, −5.01933729878958059436704696450, −3.64775196567623296293702466059, −3.13977889568755128130200501785, −2.02164885320718285635429006802, −1.10481531739259597429201642943,
1.10481531739259597429201642943, 2.02164885320718285635429006802, 3.13977889568755128130200501785, 3.64775196567623296293702466059, 5.01933729878958059436704696450, 6.43580847295372532944340458860, 7.37377737977104503172954326955, 8.011150810699143959153205853568, 8.550755747648764237970653553064, 9.292609418732479150018436473094
