| L(s) = 1 | + 19.1·5-s + 7·7-s − 16.4·11-s + 56.3·13-s + 74.1·17-s + 7.29·19-s − 105.·23-s + 240.·25-s + 130.·29-s − 113.·31-s + 133.·35-s + 279.·37-s − 298.·41-s − 122.·43-s + 528.·47-s + 49·49-s + 196.·53-s − 314.·55-s − 105.·59-s − 782.·61-s + 1.07e3·65-s − 241.·67-s + 314.·71-s − 1.07e3·73-s − 115.·77-s − 904.·79-s + 413.·83-s + ⋯ |
| L(s) = 1 | + 1.70·5-s + 0.377·7-s − 0.451·11-s + 1.20·13-s + 1.05·17-s + 0.0881·19-s − 0.954·23-s + 1.92·25-s + 0.833·29-s − 0.654·31-s + 0.646·35-s + 1.24·37-s − 1.13·41-s − 0.434·43-s + 1.64·47-s + 0.142·49-s + 0.508·53-s − 0.771·55-s − 0.233·59-s − 1.64·61-s + 2.05·65-s − 0.439·67-s + 0.526·71-s − 1.72·73-s − 0.170·77-s − 1.28·79-s + 0.547·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.269840601\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.269840601\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - 7T \) |
| good | 5 | \( 1 - 19.1T + 125T^{2} \) |
| 11 | \( 1 + 16.4T + 1.33e3T^{2} \) |
| 13 | \( 1 - 56.3T + 2.19e3T^{2} \) |
| 17 | \( 1 - 74.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 7.29T + 6.85e3T^{2} \) |
| 23 | \( 1 + 105.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 130.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 113.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 279.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 298.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 122.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 528.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 196.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 105.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 782.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 241.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 314.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.07e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 904.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 413.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 277.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.71e3T + 9.12e5T^{2} \) |
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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
−10.12314782878497013882938204266, −9.138001523295927658066898669782, −8.381337177154177576052447326778, −7.33725141794314642618668959035, −6.02262729004888889369827106814, −5.81112453785515619183167104977, −4.66081579963604752108323511304, −3.24122703840540936600014597838, −2.06640548742801182426876851387, −1.12698835611968161263588493394,
1.12698835611968161263588493394, 2.06640548742801182426876851387, 3.24122703840540936600014597838, 4.66081579963604752108323511304, 5.81112453785515619183167104977, 6.02262729004888889369827106814, 7.33725141794314642618668959035, 8.381337177154177576052447326778, 9.138001523295927658066898669782, 10.12314782878497013882938204266
