| L(s) = 1 | − 4.21·2-s − 3·3-s + 9.78·4-s + 17.5·5-s + 12.6·6-s + 14.8·7-s − 7.53·8-s + 9·9-s − 74.0·10-s − 18.9·11-s − 29.3·12-s + 31.7·13-s − 62.8·14-s − 52.6·15-s − 46.5·16-s − 84.1·17-s − 37.9·18-s + 126.·19-s + 171.·20-s − 44.6·21-s + 79.7·22-s − 21.1·23-s + 22.6·24-s + 183.·25-s − 133.·26-s − 27·27-s + 145.·28-s + ⋯ |
| L(s) = 1 | − 1.49·2-s − 0.577·3-s + 1.22·4-s + 1.57·5-s + 0.860·6-s + 0.804·7-s − 0.333·8-s + 0.333·9-s − 2.34·10-s − 0.518·11-s − 0.706·12-s + 0.677·13-s − 1.19·14-s − 0.906·15-s − 0.726·16-s − 1.20·17-s − 0.497·18-s + 1.52·19-s + 1.92·20-s − 0.464·21-s + 0.772·22-s − 0.191·23-s + 0.192·24-s + 1.46·25-s − 1.01·26-s − 0.192·27-s + 0.984·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.9744982040\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9744982040\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 59 | \( 1 + 59T \) |
| good | 2 | \( 1 + 4.21T + 8T^{2} \) |
| 5 | \( 1 - 17.5T + 125T^{2} \) |
| 7 | \( 1 - 14.8T + 343T^{2} \) |
| 11 | \( 1 + 18.9T + 1.33e3T^{2} \) |
| 13 | \( 1 - 31.7T + 2.19e3T^{2} \) |
| 17 | \( 1 + 84.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 126.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 21.1T + 1.21e4T^{2} \) |
| 29 | \( 1 + 12.6T + 2.43e4T^{2} \) |
| 31 | \( 1 - 215.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 3.62T + 5.06e4T^{2} \) |
| 41 | \( 1 + 381.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 168.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 613.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 270.T + 1.48e5T^{2} \) |
| 61 | \( 1 - 311.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 375.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 987.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 359.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 933.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.26e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 273.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 897.T + 9.12e5T^{2} \) |
| 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
−11.78605938195553725856045506059, −10.84133969463592363413002744034, −10.17585446946839226603787604101, −9.303942910179749152973314130608, −8.392637150619768251435758109005, −7.16462251050260520026012806076, −6.03855946941561769933655651333, −4.91729102188014468784388875909, −2.20894023297450445440531222765, −1.07232283918971311365301094771,
1.07232283918971311365301094771, 2.20894023297450445440531222765, 4.91729102188014468784388875909, 6.03855946941561769933655651333, 7.16462251050260520026012806076, 8.392637150619768251435758109005, 9.303942910179749152973314130608, 10.17585446946839226603787604101, 10.84133969463592363413002744034, 11.78605938195553725856045506059
