| L(s) = 1 | − 2·2-s + 7.70·3-s + 4·4-s − 5·5-s − 15.4·6-s + 17.8·7-s − 8·8-s + 32.4·9-s + 10·10-s + 38.1·11-s + 30.8·12-s − 13·13-s − 35.7·14-s − 38.5·15-s + 16·16-s + 69.7·17-s − 64.8·18-s − 152.·19-s − 20·20-s + 137.·21-s − 76.3·22-s + 167.·23-s − 61.6·24-s + 25·25-s + 26·26-s + 41.7·27-s + 71.5·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.48·3-s + 0.5·4-s − 0.447·5-s − 1.04·6-s + 0.965·7-s − 0.353·8-s + 1.20·9-s + 0.316·10-s + 1.04·11-s + 0.741·12-s − 0.277·13-s − 0.682·14-s − 0.663·15-s + 0.250·16-s + 0.995·17-s − 0.848·18-s − 1.84·19-s − 0.223·20-s + 1.43·21-s − 0.740·22-s + 1.51·23-s − 0.524·24-s + 0.200·25-s + 0.196·26-s + 0.297·27-s + 0.482·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 130 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.014636067\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.014636067\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 5T \) |
| 13 | \( 1 + 13T \) |
| good | 3 | \( 1 - 7.70T + 27T^{2} \) |
| 7 | \( 1 - 17.8T + 343T^{2} \) |
| 11 | \( 1 - 38.1T + 1.33e3T^{2} \) |
| 17 | \( 1 - 69.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + 152.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 167.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 187.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 91.7T + 2.97e4T^{2} \) |
| 37 | \( 1 + 127.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 239.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 203.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 333.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 585.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 568.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 278.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 617.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 319.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 241.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 60.4T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.20e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 557.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 241.T + 9.12e5T^{2} \) |
| 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
−12.83276631086313960662308754031, −11.71440749110991913760591008001, −10.60199092606174786114806412270, −9.354646774790104184859736613312, −8.507688877116311488866153316027, −7.911299962931347322462937407290, −6.70621648449117289917079969740, −4.49796171678336223183279309291, −3.05089722318831834150428472153, −1.53334450030353088230815352450,
1.53334450030353088230815352450, 3.05089722318831834150428472153, 4.49796171678336223183279309291, 6.70621648449117289917079969740, 7.911299962931347322462937407290, 8.507688877116311488866153316027, 9.354646774790104184859736613312, 10.60199092606174786114806412270, 11.71440749110991913760591008001, 12.83276631086313960662308754031
