| L(s) = 1 | + (0.535 − 0.844i)2-s + (2.45 − 2.30i)3-s + (−0.425 − 0.904i)4-s + (0.826 + 2.07i)5-s + (−0.630 − 3.30i)6-s + (−1.48 + 1.07i)7-s + (−0.992 − 0.125i)8-s + (0.522 − 8.30i)9-s + (2.19 + 0.414i)10-s + (−1.26 + 1.99i)11-s + (−3.12 − 1.23i)12-s + (−0.224 + 3.56i)13-s + (0.115 + 1.82i)14-s + (6.81 + 3.19i)15-s + (−0.637 + 0.770i)16-s + (0.186 − 0.395i)17-s + ⋯ |
| L(s) = 1 | + (0.378 − 0.597i)2-s + (1.41 − 1.33i)3-s + (−0.212 − 0.452i)4-s + (0.369 + 0.929i)5-s + (−0.257 − 1.34i)6-s + (−0.560 + 0.406i)7-s + (−0.350 − 0.0443i)8-s + (0.174 − 2.76i)9-s + (0.694 + 0.131i)10-s + (−0.380 + 0.600i)11-s + (−0.903 − 0.357i)12-s + (−0.0621 + 0.988i)13-s + (0.0307 + 0.488i)14-s + (1.75 + 0.824i)15-s + (−0.159 + 0.192i)16-s + (0.0451 − 0.0959i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 250 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0808 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 250 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0808 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.57379 - 1.45135i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.57379 - 1.45135i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.535 + 0.844i)T \) |
| 5 | \( 1 + (-0.826 - 2.07i)T \) |
| good | 3 | \( 1 + (-2.45 + 2.30i)T + (0.188 - 2.99i)T^{2} \) |
| 7 | \( 1 + (1.48 - 1.07i)T + (2.16 - 6.65i)T^{2} \) |
| 11 | \( 1 + (1.26 - 1.99i)T + (-4.68 - 9.95i)T^{2} \) |
| 13 | \( 1 + (0.224 - 3.56i)T + (-12.8 - 1.62i)T^{2} \) |
| 17 | \( 1 + (-0.186 + 0.395i)T + (-10.8 - 13.0i)T^{2} \) |
| 19 | \( 1 + (-0.151 - 0.142i)T + (1.19 + 18.9i)T^{2} \) |
| 23 | \( 1 + (3.49 + 0.897i)T + (20.1 + 11.0i)T^{2} \) |
| 29 | \( 1 + (0.0825 + 0.0453i)T + (15.5 + 24.4i)T^{2} \) |
| 31 | \( 1 + (2.88 - 6.14i)T + (-19.7 - 23.8i)T^{2} \) |
| 37 | \( 1 + (-7.52 + 9.09i)T + (-6.93 - 36.3i)T^{2} \) |
| 41 | \( 1 + (-10.9 + 2.81i)T + (35.9 - 19.7i)T^{2} \) |
| 43 | \( 1 + (-2.87 + 8.85i)T + (-34.7 - 25.2i)T^{2} \) |
| 47 | \( 1 + (8.41 - 1.06i)T + (45.5 - 11.6i)T^{2} \) |
| 53 | \( 1 + (1.09 - 5.71i)T + (-49.2 - 19.5i)T^{2} \) |
| 59 | \( 1 + (10.7 + 4.23i)T + (43.0 + 40.3i)T^{2} \) |
| 61 | \( 1 + (-3.76 - 0.965i)T + (53.4 + 29.3i)T^{2} \) |
| 67 | \( 1 + (-3.70 + 2.03i)T + (35.9 - 56.5i)T^{2} \) |
| 71 | \( 1 + (-11.4 + 1.44i)T + (68.7 - 17.6i)T^{2} \) |
| 73 | \( 1 + (-4.76 + 1.88i)T + (53.2 - 49.9i)T^{2} \) |
| 79 | \( 1 + (-2.41 + 2.26i)T + (4.96 - 78.8i)T^{2} \) |
| 83 | \( 1 + (-0.0752 - 0.0707i)T + (5.21 + 82.8i)T^{2} \) |
| 89 | \( 1 + (9.61 - 3.80i)T + (64.8 - 60.9i)T^{2} \) |
| 97 | \( 1 + (-4.05 - 2.22i)T + (51.9 + 81.8i)T^{2} \) |
| 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
−12.33792810135581309769299037419, −11.01085541109188371926640992363, −9.624303858439091445171827327696, −9.161070814777289106991146495728, −7.76241627034071985956617613800, −6.89857774261631259911454333218, −6.03671155705003759021627015751, −3.83910406585277834121378646820, −2.68584750485041562570083946944, −1.96960227925972131157008950052,
2.78609999122456405299562752522, 3.88180877388409994467728838513, 4.84811527744582757827091296270, 5.91075269160879200377356184183, 7.926447370539839214400102500139, 8.213299984715721995857360544047, 9.513540868375289933970298659347, 9.842198166107884917326617204026, 11.09956428909632513725509634185, 12.96243708950608200507020510480
