| L(s) = 1 | + (−0.735 + 0.197i)2-s + (4.42 − 16.5i)3-s + (−110. + 63.7i)4-s + (−184. − 209. i)5-s + 13.0i·6-s + (746. + 516. i)7-s + (137. − 137. i)8-s + (1.64e3 + 947. i)9-s + (177. + 117. i)10-s + (167. + 290. i)11-s + (563. + 2.10e3i)12-s + (1.50e3 + 1.50e3i)13-s + (−650. − 232. i)14-s + (−4.28e3 + 2.12e3i)15-s + (8.08e3 − 1.39e4i)16-s + (3.77e4 + 1.01e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.0650 + 0.0174i)2-s + (0.0946 − 0.353i)3-s + (−0.862 + 0.497i)4-s + (−0.661 − 0.750i)5-s + 0.0246i·6-s + (0.822 + 0.568i)7-s + (0.0950 − 0.0950i)8-s + (0.750 + 0.433i)9-s + (0.0560 + 0.0372i)10-s + (0.0379 + 0.0657i)11-s + (0.0941 + 0.351i)12-s + (0.189 + 0.189i)13-s + (−0.0633 − 0.0226i)14-s + (−0.327 + 0.162i)15-s + (0.493 − 0.854i)16-s + (1.86 + 0.498i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.855 - 0.517i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.855 - 0.517i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.37593 + 0.383767i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.37593 + 0.383767i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (184. + 209. i)T \) |
| 7 | \( 1 + (-746. - 516. i)T \) |
| good | 2 | \( 1 + (0.735 - 0.197i)T + (110. - 64i)T^{2} \) |
| 3 | \( 1 + (-4.42 + 16.5i)T + (-1.89e3 - 1.09e3i)T^{2} \) |
| 11 | \( 1 + (-167. - 290. i)T + (-9.74e6 + 1.68e7i)T^{2} \) |
| 13 | \( 1 + (-1.50e3 - 1.50e3i)T + 6.27e7iT^{2} \) |
| 17 | \( 1 + (-3.77e4 - 1.01e4i)T + (3.55e8 + 2.05e8i)T^{2} \) |
| 19 | \( 1 + (1.54e4 - 2.67e4i)T + (-4.46e8 - 7.74e8i)T^{2} \) |
| 23 | \( 1 + (-1.41e4 - 5.28e4i)T + (-2.94e9 + 1.70e9i)T^{2} \) |
| 29 | \( 1 + 1.45e5iT - 1.72e10T^{2} \) |
| 31 | \( 1 + (-1.15e5 + 6.66e4i)T + (1.37e10 - 2.38e10i)T^{2} \) |
| 37 | \( 1 + (4.01e5 - 1.07e5i)T + (8.22e10 - 4.74e10i)T^{2} \) |
| 41 | \( 1 - 4.85e5iT - 1.94e11T^{2} \) |
| 43 | \( 1 + (-2.86e4 + 2.86e4i)T - 2.71e11iT^{2} \) |
| 47 | \( 1 + (1.20e5 + 4.48e5i)T + (-4.38e11 + 2.53e11i)T^{2} \) |
| 53 | \( 1 + (5.83e4 + 1.56e4i)T + (1.01e12 + 5.87e11i)T^{2} \) |
| 59 | \( 1 + (-4.14e5 - 7.18e5i)T + (-1.24e12 + 2.15e12i)T^{2} \) |
| 61 | \( 1 + (-2.22e6 - 1.28e6i)T + (1.57e12 + 2.72e12i)T^{2} \) |
| 67 | \( 1 + (-6.22e5 + 2.32e6i)T + (-5.24e12 - 3.03e12i)T^{2} \) |
| 71 | \( 1 - 5.00e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + (1.44e4 - 5.40e4i)T + (-9.56e12 - 5.52e12i)T^{2} \) |
| 79 | \( 1 + (4.59e6 + 2.65e6i)T + (9.60e12 + 1.66e13i)T^{2} \) |
| 83 | \( 1 + (-3.45e6 - 3.45e6i)T + 2.71e13iT^{2} \) |
| 89 | \( 1 + (3.44e6 - 5.97e6i)T + (-2.21e13 - 3.83e13i)T^{2} \) |
| 97 | \( 1 + (1.51e5 - 1.51e5i)T - 8.07e13iT^{2} \) |
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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
−15.08045946112459191020327686111, −13.74986725287608407518794992576, −12.59483147484728600266562297801, −11.84480486164355007911759150049, −9.860049798745633330753126477355, −8.320131048940401402827663076506, −7.78743284462070727432514078468, −5.25325950619721082933601496508, −3.93349566428778909817942604506, −1.30905465010967130635749420100,
0.853225570519997417949230620325, 3.66569340561732480687862008279, 4.95065248590208996676848731877, 7.05791511157009843653851182999, 8.482431722433597644174428146495, 10.03643834133512652469708249144, 10.86597095711118132871313982258, 12.46743820658904890583286580587, 14.11587917012014724292681010811, 14.69423312989552836619034560364
