| L(s) = 1 | + 2·2-s + 7.35·3-s + 4·4-s + 14.7·6-s + 8·8-s + 27.0·9-s + 57.0·11-s + 29.4·12-s + 66.4·13-s + 16·16-s + 58.5·17-s + 54.0·18-s − 15.0·19-s + 114.·22-s + 143.·23-s + 58.8·24-s + 132.·26-s + 0.299·27-s − 17.0·29-s − 139.·31-s + 32·32-s + 419.·33-s + 117.·34-s + 108.·36-s − 227.·37-s − 30.0·38-s + 488.·39-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.41·3-s + 0.5·4-s + 1.00·6-s + 0.353·8-s + 1.00·9-s + 1.56·11-s + 0.707·12-s + 1.41·13-s + 0.250·16-s + 0.834·17-s + 0.708·18-s − 0.181·19-s + 1.10·22-s + 1.29·23-s + 0.500·24-s + 1.00·26-s + 0.00213·27-s − 0.109·29-s − 0.810·31-s + 0.176·32-s + 2.21·33-s + 0.590·34-s + 0.500·36-s − 1.01·37-s − 0.128·38-s + 2.00·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(8.132615349\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.132615349\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 7.35T + 27T^{2} \) |
| 11 | \( 1 - 57.0T + 1.33e3T^{2} \) |
| 13 | \( 1 - 66.4T + 2.19e3T^{2} \) |
| 17 | \( 1 - 58.5T + 4.91e3T^{2} \) |
| 19 | \( 1 + 15.0T + 6.85e3T^{2} \) |
| 23 | \( 1 - 143.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 17.0T + 2.43e4T^{2} \) |
| 31 | \( 1 + 139.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 227.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 37.3T + 6.89e4T^{2} \) |
| 43 | \( 1 + 147.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 161.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 542.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 903.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 505.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 729.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 36.7T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.14e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 243.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 818.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 300.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.72e3T + 9.12e5T^{2} \) |
| 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
−8.632635695534886919606667818635, −7.914782780551181445176415310019, −6.98766798035700847350752611049, −6.37956139682561809185018902623, −5.41465773450357956527299587511, −4.30366159962744882513138716330, −3.46526090139096670713161458933, −3.25490459458553564010547388473, −1.87214192974584680673631806318, −1.19711399755622482540963679354,
1.19711399755622482540963679354, 1.87214192974584680673631806318, 3.25490459458553564010547388473, 3.46526090139096670713161458933, 4.30366159962744882513138716330, 5.41465773450357956527299587511, 6.37956139682561809185018902623, 6.98766798035700847350752611049, 7.914782780551181445176415310019, 8.632635695534886919606667818635
