| L(s) = 1 | + (−0.684 + 1.87i)2-s + (−1.59 + 0.581i)3-s + (−3.06 − 2.57i)4-s + (19.8 + 3.49i)5-s − 3.40i·6-s + (1.06 − 6.03i)7-s + (6.92 − 4.00i)8-s + (−18.4 + 15.4i)9-s + (−20.1 + 34.9i)10-s + (29.9 + 51.8i)11-s + (6.39 + 2.32i)12-s + (−10.7 + 12.8i)13-s + (10.6 + 6.12i)14-s + (−33.7 + 5.94i)15-s + (2.77 + 15.7i)16-s + (65.2 + 77.7i)17-s + ⋯ |
| L(s) = 1 | + (−0.241 + 0.664i)2-s + (−0.307 + 0.111i)3-s + (−0.383 − 0.321i)4-s + (1.77 + 0.312i)5-s − 0.231i·6-s + (0.0574 − 0.325i)7-s + (0.306 − 0.176i)8-s + (−0.684 + 0.573i)9-s + (−0.637 + 1.10i)10-s + (0.819 + 1.42i)11-s + (0.153 + 0.0559i)12-s + (−0.229 + 0.273i)13-s + (0.202 + 0.116i)14-s + (−0.580 + 0.102i)15-s + (0.0434 + 0.246i)16-s + (0.931 + 1.10i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.139 - 0.990i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.139 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.09444 + 0.951195i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.09444 + 0.951195i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.684 - 1.87i)T \) |
| 37 | \( 1 + (191. - 117. i)T \) |
| good | 3 | \( 1 + (1.59 - 0.581i)T + (20.6 - 17.3i)T^{2} \) |
| 5 | \( 1 + (-19.8 - 3.49i)T + (117. + 42.7i)T^{2} \) |
| 7 | \( 1 + (-1.06 + 6.03i)T + (-322. - 117. i)T^{2} \) |
| 11 | \( 1 + (-29.9 - 51.8i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + (10.7 - 12.8i)T + (-381. - 2.16e3i)T^{2} \) |
| 17 | \( 1 + (-65.2 - 77.7i)T + (-853. + 4.83e3i)T^{2} \) |
| 19 | \( 1 + (39.0 + 107. i)T + (-5.25e3 + 4.40e3i)T^{2} \) |
| 23 | \( 1 + (74.5 + 43.0i)T + (6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-189. + 109. i)T + (1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 - 5.57iT - 2.97e4T^{2} \) |
| 41 | \( 1 + (211. + 177. i)T + (1.19e4 + 6.78e4i)T^{2} \) |
| 43 | \( 1 + 336. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + (-30.7 + 53.1i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (62.2 + 353. i)T + (-1.39e5 + 5.09e4i)T^{2} \) |
| 59 | \( 1 + (394. - 69.5i)T + (1.92e5 - 7.02e4i)T^{2} \) |
| 61 | \( 1 + (151. - 180. i)T + (-3.94e4 - 2.23e5i)T^{2} \) |
| 67 | \( 1 + (-120. + 684. i)T + (-2.82e5 - 1.02e5i)T^{2} \) |
| 71 | \( 1 + (408. - 148. i)T + (2.74e5 - 2.30e5i)T^{2} \) |
| 73 | \( 1 - 627.T + 3.89e5T^{2} \) |
| 79 | \( 1 + (985. + 173. i)T + (4.63e5 + 1.68e5i)T^{2} \) |
| 83 | \( 1 + (-503. + 422. i)T + (9.92e4 - 5.63e5i)T^{2} \) |
| 89 | \( 1 + (704. - 124. i)T + (6.62e5 - 2.41e5i)T^{2} \) |
| 97 | \( 1 + (79.6 + 45.9i)T + (4.56e5 + 7.90e5i)T^{2} \) |
| 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
−14.24832278070390707287051536003, −13.65714387028550998687479199997, −12.26317158367885192807552156274, −10.50208351940244116844402695483, −9.914474625171291621472201029251, −8.699550772356033126179405931001, −6.96411357027129659445810283003, −6.03995376285411876700500096069, −4.79450191744293411025719195794, −1.99187341388958873459643793601,
1.22241872347172043225706111614, 3.04772004664454862887471278408, 5.45454674601883102093954602751, 6.23012900780805487036823393786, 8.512332108332028607697596799657, 9.395671246589121235993882177605, 10.36033826199202067202732558528, 11.70689251424503346207330412005, 12.54565643239685692691553033737, 13.98459561107404793953174278180
