| L(s) = 1 | + (−0.495 + 0.132i)2-s + (15.4 + 4.13i)3-s + (−13.6 + 7.86i)4-s + (12.6 + 21.5i)5-s − 8.18·6-s + (−36.5 − 32.6i)7-s + (11.5 − 11.5i)8-s + (150. + 87.0i)9-s + (−9.11 − 9.00i)10-s + (42.7 + 74.0i)11-s + (−242. + 65.0i)12-s + (131. − 131. i)13-s + (22.4 + 11.3i)14-s + (105. + 385. i)15-s + (121. − 210. i)16-s + (49.9 − 186. i)17-s + ⋯ |
| L(s) = 1 | + (−0.123 + 0.0331i)2-s + (1.71 + 0.459i)3-s + (−0.851 + 0.491i)4-s + (0.505 + 0.862i)5-s − 0.227·6-s + (−0.745 − 0.667i)7-s + (0.179 − 0.179i)8-s + (1.86 + 1.07i)9-s + (−0.0911 − 0.0900i)10-s + (0.353 + 0.611i)11-s + (−1.68 + 0.451i)12-s + (0.780 − 0.780i)13-s + (0.114 + 0.0578i)14-s + (0.470 + 1.71i)15-s + (0.475 − 0.823i)16-s + (0.172 − 0.645i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.589 - 0.807i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.589 - 0.807i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.71473 + 0.871314i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.71473 + 0.871314i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-12.6 - 21.5i)T \) |
| 7 | \( 1 + (36.5 + 32.6i)T \) |
| good | 2 | \( 1 + (0.495 - 0.132i)T + (13.8 - 8i)T^{2} \) |
| 3 | \( 1 + (-15.4 - 4.13i)T + (70.1 + 40.5i)T^{2} \) |
| 11 | \( 1 + (-42.7 - 74.0i)T + (-7.32e3 + 1.26e4i)T^{2} \) |
| 13 | \( 1 + (-131. + 131. i)T - 2.85e4iT^{2} \) |
| 17 | \( 1 + (-49.9 + 186. i)T + (-7.23e4 - 4.17e4i)T^{2} \) |
| 19 | \( 1 + (435. + 251. i)T + (6.51e4 + 1.12e5i)T^{2} \) |
| 23 | \( 1 + (58.9 + 220. i)T + (-2.42e5 + 1.39e5i)T^{2} \) |
| 29 | \( 1 + 592. iT - 7.07e5T^{2} \) |
| 31 | \( 1 + (-244. - 423. i)T + (-4.61e5 + 7.99e5i)T^{2} \) |
| 37 | \( 1 + (-75.9 + 20.3i)T + (1.62e6 - 9.37e5i)T^{2} \) |
| 41 | \( 1 + 1.39e3T + 2.82e6T^{2} \) |
| 43 | \( 1 + (838. - 838. i)T - 3.41e6iT^{2} \) |
| 47 | \( 1 + (-1.80e3 + 483. i)T + (4.22e6 - 2.43e6i)T^{2} \) |
| 53 | \( 1 + (-476. - 127. i)T + (6.83e6 + 3.94e6i)T^{2} \) |
| 59 | \( 1 + (-704. + 406. i)T + (6.05e6 - 1.04e7i)T^{2} \) |
| 61 | \( 1 + (-1.16e3 + 2.02e3i)T + (-6.92e6 - 1.19e7i)T^{2} \) |
| 67 | \( 1 + (1.80e3 - 6.74e3i)T + (-1.74e7 - 1.00e7i)T^{2} \) |
| 71 | \( 1 + 767.T + 2.54e7T^{2} \) |
| 73 | \( 1 + (-2.62e3 - 702. i)T + (2.45e7 + 1.41e7i)T^{2} \) |
| 79 | \( 1 + (4.38e3 + 2.53e3i)T + (1.94e7 + 3.37e7i)T^{2} \) |
| 83 | \( 1 + (7.57e3 - 7.57e3i)T - 4.74e7iT^{2} \) |
| 89 | \( 1 + (1.21e4 + 7.03e3i)T + (3.13e7 + 5.43e7i)T^{2} \) |
| 97 | \( 1 + (-6.31e3 - 6.31e3i)T + 8.85e7iT^{2} \) |
| 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
−15.66867529254414544269825114841, −14.58417561161625524465922685825, −13.65351071837540927000344706793, −13.01585287611130519075449918449, −10.34713291314429267620072196290, −9.561746600537280689492345006806, −8.411768471965796285403167889135, −7.05781655285589868160827531458, −4.09980225693804981323627702417, −2.91381481953917369125390469952,
1.66816617472115531037701656227, 3.86508462909133345089357202214, 6.12314149344513636220403180227, 8.522905463946453008904947432832, 8.849640809788494291971636906854, 9.933991675535276569673890280166, 12.56967295718050740437029998393, 13.40477217619200372705490025617, 14.15450702261778387101133005753, 15.28770446022547749536128437571
