| L(s) = 1 | + (−6.43 − 14.6i)2-s + (−40.5 − 23.3i)3-s + (−173. + 188. i)4-s + (478. + 829. i)5-s + (−81.7 + 743. i)6-s + (544. − 2.33e3i)7-s + (3.87e3 + 1.32e3i)8-s + (1.09e3 + 1.89e3i)9-s + (9.06e3 − 1.23e4i)10-s + (−1.29e4 − 7.48e3i)11-s + (1.14e4 − 3.58e3i)12-s + 2.82e4·13-s + (−3.77e4 + 7.07e3i)14-s − 4.47e4i·15-s + (−5.59e3 − 6.52e4i)16-s + (−3.03e4 + 5.25e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.402 − 0.915i)2-s + (−0.5 − 0.288i)3-s + (−0.676 + 0.736i)4-s + (0.765 + 1.32i)5-s + (−0.0631 + 0.573i)6-s + (0.226 − 0.973i)7-s + (0.946 + 0.322i)8-s + (0.166 + 0.288i)9-s + (0.906 − 1.23i)10-s + (−0.885 − 0.511i)11-s + (0.550 − 0.173i)12-s + 0.989·13-s + (−0.982 + 0.184i)14-s − 0.884i·15-s + (−0.0853 − 0.996i)16-s + (−0.363 + 0.629i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.489 - 0.872i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.489 - 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.715876 + 0.419188i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.715876 + 0.419188i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (6.43 + 14.6i)T \) |
| 3 | \( 1 + (40.5 + 23.3i)T \) |
| 7 | \( 1 + (-544. + 2.33e3i)T \) |
| good | 5 | \( 1 + (-478. - 829. i)T + (-1.95e5 + 3.38e5i)T^{2} \) |
| 11 | \( 1 + (1.29e4 + 7.48e3i)T + (1.07e8 + 1.85e8i)T^{2} \) |
| 13 | \( 1 - 2.82e4T + 8.15e8T^{2} \) |
| 17 | \( 1 + (3.03e4 - 5.25e4i)T + (-3.48e9 - 6.04e9i)T^{2} \) |
| 19 | \( 1 + (9.51e4 - 5.49e4i)T + (8.49e9 - 1.47e10i)T^{2} \) |
| 23 | \( 1 + (-3.46e4 + 1.99e4i)T + (3.91e10 - 6.78e10i)T^{2} \) |
| 29 | \( 1 + 5.68e5T + 5.00e11T^{2} \) |
| 31 | \( 1 + (-6.15e5 - 3.55e5i)T + (4.26e11 + 7.38e11i)T^{2} \) |
| 37 | \( 1 + (4.13e5 + 7.17e5i)T + (-1.75e12 + 3.04e12i)T^{2} \) |
| 41 | \( 1 - 4.10e6T + 7.98e12T^{2} \) |
| 43 | \( 1 + 1.55e6iT - 1.16e13T^{2} \) |
| 47 | \( 1 + (6.70e6 - 3.86e6i)T + (1.19e13 - 2.06e13i)T^{2} \) |
| 53 | \( 1 + (4.86e6 - 8.42e6i)T + (-3.11e13 - 5.39e13i)T^{2} \) |
| 59 | \( 1 + (-1.60e7 - 9.25e6i)T + (7.34e13 + 1.27e14i)T^{2} \) |
| 61 | \( 1 + (-1.34e7 - 2.33e7i)T + (-9.58e13 + 1.66e14i)T^{2} \) |
| 67 | \( 1 + (1.88e7 + 1.08e7i)T + (2.03e14 + 3.51e14i)T^{2} \) |
| 71 | \( 1 - 3.99e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 + (1.61e7 - 2.79e7i)T + (-4.03e14 - 6.98e14i)T^{2} \) |
| 79 | \( 1 + (8.23e6 - 4.75e6i)T + (7.58e14 - 1.31e15i)T^{2} \) |
| 83 | \( 1 - 8.38e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 + (4.33e7 + 7.51e7i)T + (-1.96e15 + 3.40e15i)T^{2} \) |
| 97 | \( 1 + 6.40e7T + 7.83e15T^{2} \) |
| 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
−12.91046882631934074708703803696, −11.17549422404218764766894587935, −10.77489189551922191310154368247, −10.05274942933243073361686757739, −8.352452738764323873129863305742, −7.13657331876362563037111893941, −5.88029933610565175584298474223, −3.99582147349411454069359269354, −2.61361687756231983290924540935, −1.27793889553119942588684998699,
0.32526953019685428788243766472, 1.81328882278433163132881495297, 4.68502099822535713303801280430, 5.36628413224943534351049180037, 6.35748353967166615385474771270, 8.145214073845245978144207090622, 9.026467842422352184242438626855, 9.828100461719090484131603483050, 11.23060528699274960342178503334, 12.78920439320517741358512182242
