| L(s) = 1 | + (−7.78 + 2.28i)2-s + (10.2 + 11.7i)3-s + (1.49 − 0.961i)4-s + (−172. + 24.8i)5-s + (−106. − 68.3i)6-s + (−306. − 139. i)7-s + (330. − 381. i)8-s + (−34.5 + 240. i)9-s + (1.28e3 − 587. i)10-s + (−121. + 413. i)11-s + (26.6 + 7.81i)12-s + (−640. − 1.40e3i)13-s + (2.70e3 + 388. i)14-s + (−2.05e3 − 1.78e3i)15-s + (−1.74e3 + 3.82e3i)16-s + (−2.18e3 + 3.40e3i)17-s + ⋯ |
| L(s) = 1 | + (−0.972 + 0.285i)2-s + (0.378 + 0.436i)3-s + (0.0233 − 0.0150i)4-s + (−1.38 + 0.198i)5-s + (−0.492 − 0.316i)6-s + (−0.893 − 0.407i)7-s + (0.645 − 0.744i)8-s + (−0.0474 + 0.329i)9-s + (1.28 − 0.587i)10-s + (−0.0911 + 0.310i)11-s + (0.0153 + 0.00452i)12-s + (−0.291 − 0.637i)13-s + (0.985 + 0.141i)14-s + (−0.609 − 0.527i)15-s + (−0.426 + 0.934i)16-s + (−0.445 + 0.692i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0331i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.999 - 0.0331i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.487885 + 0.00808305i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.487885 + 0.00808305i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-10.2 - 11.7i)T \) |
| 23 | \( 1 + (-9.58e3 - 7.49e3i)T \) |
| good | 2 | \( 1 + (7.78 - 2.28i)T + (53.8 - 34.6i)T^{2} \) |
| 5 | \( 1 + (172. - 24.8i)T + (1.49e4 - 4.40e3i)T^{2} \) |
| 7 | \( 1 + (306. + 139. i)T + (7.70e4 + 8.89e4i)T^{2} \) |
| 11 | \( 1 + (121. - 413. i)T + (-1.49e6 - 9.57e5i)T^{2} \) |
| 13 | \( 1 + (640. + 1.40e3i)T + (-3.16e6 + 3.64e6i)T^{2} \) |
| 17 | \( 1 + (2.18e3 - 3.40e3i)T + (-1.00e7 - 2.19e7i)T^{2} \) |
| 19 | \( 1 + (1.50e3 + 2.34e3i)T + (-1.95e7 + 4.27e7i)T^{2} \) |
| 29 | \( 1 + (-3.20e4 - 2.06e4i)T + (2.47e8 + 5.41e8i)T^{2} \) |
| 31 | \( 1 + (-1.89e4 + 2.18e4i)T + (-1.26e8 - 8.78e8i)T^{2} \) |
| 37 | \( 1 + (-3.12e4 - 4.48e3i)T + (2.46e9 + 7.22e8i)T^{2} \) |
| 41 | \( 1 + (1.14e4 + 7.94e4i)T + (-4.55e9 + 1.33e9i)T^{2} \) |
| 43 | \( 1 + (-3.40e4 + 2.94e4i)T + (8.99e8 - 6.25e9i)T^{2} \) |
| 47 | \( 1 + 1.32e5T + 1.07e10T^{2} \) |
| 53 | \( 1 + (-2.13e5 - 9.75e4i)T + (1.45e10 + 1.67e10i)T^{2} \) |
| 59 | \( 1 + (-7.01e4 - 1.53e5i)T + (-2.76e10 + 3.18e10i)T^{2} \) |
| 61 | \( 1 + (2.97e5 + 2.57e5i)T + (7.33e9 + 5.09e10i)T^{2} \) |
| 67 | \( 1 + (5.11e4 + 1.74e5i)T + (-7.60e10 + 4.89e10i)T^{2} \) |
| 71 | \( 1 + (-3.32e5 + 9.75e4i)T + (1.07e11 - 6.92e10i)T^{2} \) |
| 73 | \( 1 + (-5.62e4 + 3.61e4i)T + (6.28e10 - 1.37e11i)T^{2} \) |
| 79 | \( 1 + (5.89e4 - 2.69e4i)T + (1.59e11 - 1.83e11i)T^{2} \) |
| 83 | \( 1 + (5.54e5 + 7.97e4i)T + (3.13e11 + 9.21e10i)T^{2} \) |
| 89 | \( 1 + (-7.82e5 + 6.78e5i)T + (7.07e10 - 4.91e11i)T^{2} \) |
| 97 | \( 1 + (5.94e5 - 8.54e4i)T + (7.99e11 - 2.34e11i)T^{2} \) |
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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
−13.46059448340022090976108461359, −12.43875599538894252398045949710, −10.87160201215329291592638293829, −9.969812010248551691140106205469, −8.786930333852331228712820881451, −7.81618574141960647837480234795, −6.88742854042575840889628908871, −4.41458326035930665258613919199, −3.30379499592137300300105914774, −0.42817974838697468563054661047,
0.73725029607638486670197656230, 2.77425167412092203729748967168, 4.53705333321667487546790892214, 6.70989946194036270080754376417, 8.011245182915015344739049327067, 8.786209146007831345546625005644, 9.851657031730180725853214851526, 11.30062970702866639988436832114, 12.17333953844959202985396465294, 13.39738922007169917543553317367
