| L(s) = 1 | + (−0.442 + 2.79i)2-s + (10.3 + 4.30i)3-s + (−7.60 − 2.47i)4-s + (19.2 − 9.81i)5-s + (−16.6 + 27.1i)6-s + (20.5 + 33.4i)7-s + (10.2 − 20.1i)8-s + (32.2 + 32.2i)9-s + (18.8 + 58.1i)10-s + (200. + 15.7i)11-s + (−68.4 − 58.4i)12-s + (−131. + 31.6i)13-s + (−102. + 42.4i)14-s + (242. − 19.0i)15-s + (51.7 + 37.6i)16-s + (43.5 + 51.0i)17-s + ⋯ |
| L(s) = 1 | + (−0.110 + 0.698i)2-s + (1.15 + 0.478i)3-s + (−0.475 − 0.154i)4-s + (0.770 − 0.392i)5-s + (−0.461 + 0.753i)6-s + (0.418 + 0.682i)7-s + (0.160 − 0.315i)8-s + (0.397 + 0.397i)9-s + (0.188 + 0.581i)10-s + (1.65 + 0.130i)11-s + (−0.475 − 0.405i)12-s + (−0.780 + 0.187i)13-s + (−0.523 + 0.216i)14-s + (1.07 − 0.0847i)15-s + (0.202 + 0.146i)16-s + (0.150 + 0.176i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 82 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.232 - 0.972i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 82 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.232 - 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.98494 + 1.56714i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.98494 + 1.56714i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.442 - 2.79i)T \) |
| 41 | \( 1 + (978. + 1.36e3i)T \) |
| good | 3 | \( 1 + (-10.3 - 4.30i)T + (57.2 + 57.2i)T^{2} \) |
| 5 | \( 1 + (-19.2 + 9.81i)T + (367. - 505. i)T^{2} \) |
| 7 | \( 1 + (-20.5 - 33.4i)T + (-1.09e3 + 2.13e3i)T^{2} \) |
| 11 | \( 1 + (-200. - 15.7i)T + (1.44e4 + 2.29e3i)T^{2} \) |
| 13 | \( 1 + (131. - 31.6i)T + (2.54e4 - 1.29e4i)T^{2} \) |
| 17 | \( 1 + (-43.5 - 51.0i)T + (-1.30e4 + 8.24e4i)T^{2} \) |
| 19 | \( 1 + (422. + 101. i)T + (1.16e5 + 5.91e4i)T^{2} \) |
| 23 | \( 1 + (-251. - 345. i)T + (-8.64e4 + 2.66e5i)T^{2} \) |
| 29 | \( 1 + (292. - 342. i)T + (-1.10e5 - 6.98e5i)T^{2} \) |
| 31 | \( 1 + (-337. + 109. i)T + (7.47e5 - 5.42e5i)T^{2} \) |
| 37 | \( 1 + (-80.7 + 248. i)T + (-1.51e6 - 1.10e6i)T^{2} \) |
| 43 | \( 1 + (-308. + 1.94e3i)T + (-3.25e6 - 1.05e6i)T^{2} \) |
| 47 | \( 1 + (-1.22e3 + 2.00e3i)T + (-2.21e6 - 4.34e6i)T^{2} \) |
| 53 | \( 1 + (3.50e3 + 2.99e3i)T + (1.23e6 + 7.79e6i)T^{2} \) |
| 59 | \( 1 + (-1.56e3 + 1.13e3i)T + (3.74e6 - 1.15e7i)T^{2} \) |
| 61 | \( 1 + (-4.99e3 + 790. i)T + (1.31e7 - 4.27e6i)T^{2} \) |
| 67 | \( 1 + (-441. - 5.61e3i)T + (-1.99e7 + 3.15e6i)T^{2} \) |
| 71 | \( 1 + (100. - 1.27e3i)T + (-2.50e7 - 3.97e6i)T^{2} \) |
| 73 | \( 1 + (3.08e3 - 3.08e3i)T - 2.83e7iT^{2} \) |
| 79 | \( 1 + (-430. + 1.04e3i)T + (-2.75e7 - 2.75e7i)T^{2} \) |
| 83 | \( 1 + 1.15e4T + 4.74e7T^{2} \) |
| 89 | \( 1 + (435. + 710. i)T + (-2.84e7 + 5.59e7i)T^{2} \) |
| 97 | \( 1 + (-2.83e3 + 223. i)T + (8.74e7 - 1.38e7i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.28849114957004785184061098389, −13.10366134511946631528105672981, −11.76825901707309573095950099882, −9.892209229304763930422651157457, −9.079095433466651221489822966785, −8.537329361631477970536777060209, −6.89130827983849906241044253274, −5.42995930788837905578395450355, −3.95762680532505137413394541639, −1.97420946274258022908721315964,
1.46930124860157088240837018565, 2.72813027978666914253324446523, 4.26906385674379850322287016994, 6.48110737419178113971243928457, 7.82570554588881066219549191000, 8.973087827365324449444940293245, 9.917692580420044477623966891820, 11.10265158942184413691702934859, 12.42389078813261617220926727196, 13.51267713694444180468201166560
