| L(s) = 1 | + 5.60·2-s + 23.4·4-s − 12.9·5-s − 18.2·7-s + 86.4·8-s − 72.7·10-s + 71.8·11-s + 31.4·13-s − 102.·14-s + 297.·16-s + 94.8·17-s − 53.7·19-s − 303.·20-s + 402.·22-s + 70.2·23-s + 43.3·25-s + 176.·26-s − 428.·28-s − 20.2·29-s − 5.95·31-s + 973.·32-s + 531.·34-s + 237.·35-s + 167.·37-s − 300.·38-s − 1.12e3·40-s − 248.·41-s + ⋯ |
| L(s) = 1 | + 1.98·2-s + 2.92·4-s − 1.16·5-s − 0.988·7-s + 3.81·8-s − 2.29·10-s + 1.96·11-s + 0.671·13-s − 1.95·14-s + 4.64·16-s + 1.35·17-s − 0.648·19-s − 3.39·20-s + 3.90·22-s + 0.636·23-s + 0.346·25-s + 1.33·26-s − 2.89·28-s − 0.129·29-s − 0.0344·31-s + 5.37·32-s + 2.68·34-s + 1.14·35-s + 0.742·37-s − 1.28·38-s − 4.43·40-s − 0.948·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(6.523181956\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.523181956\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 59 | \( 1 - 59T \) |
| good | 2 | \( 1 - 5.60T + 8T^{2} \) |
| 5 | \( 1 + 12.9T + 125T^{2} \) |
| 7 | \( 1 + 18.2T + 343T^{2} \) |
| 11 | \( 1 - 71.8T + 1.33e3T^{2} \) |
| 13 | \( 1 - 31.4T + 2.19e3T^{2} \) |
| 17 | \( 1 - 94.8T + 4.91e3T^{2} \) |
| 19 | \( 1 + 53.7T + 6.85e3T^{2} \) |
| 23 | \( 1 - 70.2T + 1.21e4T^{2} \) |
| 29 | \( 1 + 20.2T + 2.43e4T^{2} \) |
| 31 | \( 1 + 5.95T + 2.97e4T^{2} \) |
| 37 | \( 1 - 167.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 248.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 472.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 140.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 21.3T + 1.48e5T^{2} \) |
| 61 | \( 1 - 375.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 410.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 420.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 865.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 53.7T + 4.93e5T^{2} \) |
| 83 | \( 1 + 39.4T + 5.71e5T^{2} \) |
| 89 | \( 1 - 354.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.41e3T + 9.12e5T^{2} \) |
| 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
−10.95672916511583804123511882850, −9.831580009146997731019316073898, −8.364031520997691751661398971320, −7.17859345025686869385377667190, −6.57757137325178241549009037829, −5.78188320888043805355657649646, −4.42977628137158356612522821180, −3.68448066389997660378954326022, −3.20436660276941560862749670150, −1.35691811352080872006114298083,
1.35691811352080872006114298083, 3.20436660276941560862749670150, 3.68448066389997660378954326022, 4.42977628137158356612522821180, 5.78188320888043805355657649646, 6.57757137325178241549009037829, 7.17859345025686869385377667190, 8.364031520997691751661398971320, 9.831580009146997731019316073898, 10.95672916511583804123511882850
