| L(s) = 1 | + (−13.3 + 3.58i)2-s + (17.7 − 66.0i)3-s + (54.8 − 31.6i)4-s + (−217. + 175. i)5-s + 946. i·6-s + (−622. + 659. i)7-s + (632. − 632. i)8-s + (−2.15e3 − 1.24e3i)9-s + (2.28e3 − 3.12e3i)10-s + (−987. − 1.71e3i)11-s + (−1.12e3 − 4.18e3i)12-s + (2.22e3 + 2.22e3i)13-s + (5.95e3 − 1.10e4i)14-s + (7.71e3 + 1.74e4i)15-s + (−1.02e4 + 1.77e4i)16-s + (2.72e4 + 7.29e3i)17-s + ⋯ |
| L(s) = 1 | + (−1.18 + 0.316i)2-s + (0.378 − 1.41i)3-s + (0.428 − 0.247i)4-s + (−0.779 + 0.626i)5-s + 1.78i·6-s + (−0.686 + 0.727i)7-s + (0.436 − 0.436i)8-s + (−0.986 − 0.569i)9-s + (0.722 − 0.986i)10-s + (−0.223 − 0.387i)11-s + (−0.187 − 0.699i)12-s + (0.280 + 0.280i)13-s + (0.580 − 1.07i)14-s + (0.590 + 1.33i)15-s + (−0.624 + 1.08i)16-s + (1.34 + 0.359i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.940 - 0.340i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.940 - 0.340i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.714393 + 0.125403i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.714393 + 0.125403i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (217. - 175. i)T \) |
| 7 | \( 1 + (622. - 659. i)T \) |
| good | 2 | \( 1 + (13.3 - 3.58i)T + (110. - 64i)T^{2} \) |
| 3 | \( 1 + (-17.7 + 66.0i)T + (-1.89e3 - 1.09e3i)T^{2} \) |
| 11 | \( 1 + (987. + 1.71e3i)T + (-9.74e6 + 1.68e7i)T^{2} \) |
| 13 | \( 1 + (-2.22e3 - 2.22e3i)T + 6.27e7iT^{2} \) |
| 17 | \( 1 + (-2.72e4 - 7.29e3i)T + (3.55e8 + 2.05e8i)T^{2} \) |
| 19 | \( 1 + (-1.60e4 + 2.78e4i)T + (-4.46e8 - 7.74e8i)T^{2} \) |
| 23 | \( 1 + (-1.94e4 - 7.27e4i)T + (-2.94e9 + 1.70e9i)T^{2} \) |
| 29 | \( 1 - 8.51e4iT - 1.72e10T^{2} \) |
| 31 | \( 1 + (-1.95e5 + 1.13e5i)T + (1.37e10 - 2.38e10i)T^{2} \) |
| 37 | \( 1 + (-1.01e5 + 2.71e4i)T + (8.22e10 - 4.74e10i)T^{2} \) |
| 41 | \( 1 - 4.81e5iT - 1.94e11T^{2} \) |
| 43 | \( 1 + (3.91e5 - 3.91e5i)T - 2.71e11iT^{2} \) |
| 47 | \( 1 + (-2.76e5 - 1.03e6i)T + (-4.38e11 + 2.53e11i)T^{2} \) |
| 53 | \( 1 + (5.84e4 + 1.56e4i)T + (1.01e12 + 5.87e11i)T^{2} \) |
| 59 | \( 1 + (-3.36e4 - 5.83e4i)T + (-1.24e12 + 2.15e12i)T^{2} \) |
| 61 | \( 1 + (1.83e6 + 1.06e6i)T + (1.57e12 + 2.72e12i)T^{2} \) |
| 67 | \( 1 + (8.12e5 - 3.03e6i)T + (-5.24e12 - 3.03e12i)T^{2} \) |
| 71 | \( 1 - 3.72e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + (6.82e5 - 2.54e6i)T + (-9.56e12 - 5.52e12i)T^{2} \) |
| 79 | \( 1 + (-4.99e6 - 2.88e6i)T + (9.60e12 + 1.66e13i)T^{2} \) |
| 83 | \( 1 + (-3.17e6 - 3.17e6i)T + 2.71e13iT^{2} \) |
| 89 | \( 1 + (1.76e6 - 3.05e6i)T + (-2.21e13 - 3.83e13i)T^{2} \) |
| 97 | \( 1 + (-4.96e6 + 4.96e6i)T - 8.07e13iT^{2} \) |
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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
−15.32293791874118107181038878741, −13.75274613189753850576953541312, −12.64824496587941199749356003255, −11.39252738921416670630247443212, −9.636006550230200056689355126202, −8.284552919423968844136591975336, −7.49423344770734273091274898165, −6.39559135103672833528461109475, −3.03839704977867846928399239127, −1.00724002477261861016431975504,
0.64263185618305968272852530920, 3.47602093183301121904082182296, 4.85603171591916831703064914947, 7.69958303026174077241363627551, 8.819955805556770789133554811016, 9.965544731910728797253500513774, 10.49092680420414454769581463056, 12.13229257962861732379325767249, 13.94796079070649378014882335153, 15.35750572996634231973780981122
