| L(s) = 1 | − 1.49·2-s + 9.02·3-s − 5.77·4-s − 5·5-s − 13.4·6-s + 4.33·7-s + 20.5·8-s + 54.4·9-s + 7.46·10-s + 48.1·11-s − 52.1·12-s + 53.0·13-s − 6.46·14-s − 45.1·15-s + 15.5·16-s − 61.0·17-s − 81.2·18-s + 12.0·19-s + 28.8·20-s + 39.1·21-s − 71.8·22-s − 23·23-s + 185.·24-s + 25·25-s − 79.1·26-s + 247.·27-s − 25.0·28-s + ⋯ |
| L(s) = 1 | − 0.527·2-s + 1.73·3-s − 0.721·4-s − 0.447·5-s − 0.916·6-s + 0.234·7-s + 0.908·8-s + 2.01·9-s + 0.235·10-s + 1.32·11-s − 1.25·12-s + 1.13·13-s − 0.123·14-s − 0.776·15-s + 0.242·16-s − 0.870·17-s − 1.06·18-s + 0.145·19-s + 0.322·20-s + 0.406·21-s − 0.696·22-s − 0.208·23-s + 1.57·24-s + 0.200·25-s − 0.596·26-s + 1.76·27-s − 0.168·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.901774149\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.901774149\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 5T \) |
| 23 | \( 1 + 23T \) |
| good | 2 | \( 1 + 1.49T + 8T^{2} \) |
| 3 | \( 1 - 9.02T + 27T^{2} \) |
| 7 | \( 1 - 4.33T + 343T^{2} \) |
| 11 | \( 1 - 48.1T + 1.33e3T^{2} \) |
| 13 | \( 1 - 53.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + 61.0T + 4.91e3T^{2} \) |
| 19 | \( 1 - 12.0T + 6.85e3T^{2} \) |
| 29 | \( 1 - 198.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 11.9T + 2.97e4T^{2} \) |
| 37 | \( 1 + 230.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 393.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 167.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 531.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 373.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 825.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 632.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 38.7T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.10e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 446.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 642.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 292.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 273.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 959.T + 9.12e5T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.65297887250768486434708809132, −12.20821721563978274666425869405, −10.66974865539131059707915073925, −9.424593560463273885485177291877, −8.707628595550669670169618043360, −8.174882123397384072111575886134, −6.85316180382623169004549196540, −4.40349074846768530393434012649, −3.50061766623195554083395588205, −1.48234843500660029761789802229,
1.48234843500660029761789802229, 3.50061766623195554083395588205, 4.40349074846768530393434012649, 6.85316180382623169004549196540, 8.174882123397384072111575886134, 8.707628595550669670169618043360, 9.424593560463273885485177291877, 10.66974865539131059707915073925, 12.20821721563978274666425869405, 13.65297887250768486434708809132
