| L(s) = 1 | − 24.7i·5-s − 50.9i·7-s + 139.·11-s + 98.9i·13-s + 16·17-s + 559.·19-s + 814. i·23-s + 13.0·25-s − 420. i·29-s + 661. i·31-s − 1.25e3·35-s + 2.47e3i·37-s + 464·41-s − 559.·43-s + 2.44e3i·47-s + ⋯ |
| L(s) = 1 | − 0.989i·5-s − 1.03i·7-s + 1.15·11-s + 0.585i·13-s + 0.0553·17-s + 1.55·19-s + 1.53i·23-s + 0.0208·25-s − 0.500i·29-s + 0.688i·31-s − 1.02·35-s + 1.80i·37-s + 0.276·41-s − 0.302·43-s + 1.10i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(2.523595119\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.523595119\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 24.7iT - 625T^{2} \) |
| 7 | \( 1 + 50.9iT - 2.40e3T^{2} \) |
| 11 | \( 1 - 139.T + 1.46e4T^{2} \) |
| 13 | \( 1 - 98.9iT - 2.85e4T^{2} \) |
| 17 | \( 1 - 16T + 8.35e4T^{2} \) |
| 19 | \( 1 - 559.T + 1.30e5T^{2} \) |
| 23 | \( 1 - 814. iT - 2.79e5T^{2} \) |
| 29 | \( 1 + 420. iT - 7.07e5T^{2} \) |
| 31 | \( 1 - 661. iT - 9.23e5T^{2} \) |
| 37 | \( 1 - 2.47e3iT - 1.87e6T^{2} \) |
| 41 | \( 1 - 464T + 2.82e6T^{2} \) |
| 43 | \( 1 + 559.T + 3.41e6T^{2} \) |
| 47 | \( 1 - 2.44e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 - 3.19e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 - 3.63e3T + 1.21e7T^{2} \) |
| 61 | \( 1 - 4.25e3iT - 1.38e7T^{2} \) |
| 67 | \( 1 + 5.59e3T + 2.01e7T^{2} \) |
| 71 | \( 1 + 3.25e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 898T + 2.83e7T^{2} \) |
| 79 | \( 1 - 8.60e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 + 419.T + 4.74e7T^{2} \) |
| 89 | \( 1 - 7.07e3T + 6.27e7T^{2} \) |
| 97 | \( 1 + 2.36e3T + 8.85e7T^{2} \) |
| 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
−9.386305076190502688588630893804, −8.494207009364998337488943701750, −7.52512179538622566818275138005, −6.89675111036017343730154798953, −5.81880219400721396186704360350, −4.82753866541667502806060471092, −4.10323369476065649740282737796, −3.19643649703433340095535988410, −1.34549808876081211947137105274, −1.06755717346053636301011323110,
0.62362980995039627544022645706, 2.06876912991547478299364811721, 2.96057282596189563932174992366, 3.80140304344882096964392359648, 5.11929040834571180651963432325, 5.94473413549739971727819287303, 6.72981431258588655335339121235, 7.47692219917678858221238443771, 8.548367294341437145059430275903, 9.213733932157563519946194622024
