| L(s) = 1 | + (−2.33 − 4.04i)2-s + (−32.1 − 18.5i)3-s + (21.1 − 36.5i)4-s + (−189. + 109. i)5-s + 173. i·6-s − 495.·8-s + (322. + 558. i)9-s + (883. + 510. i)10-s + (168. − 292. i)11-s + (−1.35e3 + 782. i)12-s − 2.48e3i·13-s + 8.09e3·15-s + (−192. − 333. i)16-s + (5.15e3 + 2.97e3i)17-s + (1.50e3 − 2.60e3i)18-s + (−1.60e3 + 929. i)19-s + ⋯ |
| L(s) = 1 | + (−0.291 − 0.505i)2-s + (−1.18 − 0.686i)3-s + (0.329 − 0.571i)4-s + (−1.51 + 0.873i)5-s + 0.801i·6-s − 0.968·8-s + (0.442 + 0.766i)9-s + (0.883 + 0.510i)10-s + (0.126 − 0.219i)11-s + (−0.783 + 0.452i)12-s − 1.12i·13-s + 2.39·15-s + (−0.0470 − 0.0815i)16-s + (1.05 + 0.606i)17-s + (0.258 − 0.447i)18-s + (−0.234 + 0.135i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.937 - 0.347i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.937 - 0.347i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.341381 + 0.0612662i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.341381 + 0.0612662i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| good | 2 | \( 1 + (2.33 + 4.04i)T + (-32 + 55.4i)T^{2} \) |
| 3 | \( 1 + (32.1 + 18.5i)T + (364.5 + 631. i)T^{2} \) |
| 5 | \( 1 + (189. - 109. i)T + (7.81e3 - 1.35e4i)T^{2} \) |
| 11 | \( 1 + (-168. + 292. i)T + (-8.85e5 - 1.53e6i)T^{2} \) |
| 13 | \( 1 + 2.48e3iT - 4.82e6T^{2} \) |
| 17 | \( 1 + (-5.15e3 - 2.97e3i)T + (1.20e7 + 2.09e7i)T^{2} \) |
| 19 | \( 1 + (1.60e3 - 929. i)T + (2.35e7 - 4.07e7i)T^{2} \) |
| 23 | \( 1 + (-5.19e3 - 8.99e3i)T + (-7.40e7 + 1.28e8i)T^{2} \) |
| 29 | \( 1 + 137.T + 5.94e8T^{2} \) |
| 31 | \( 1 + (2.09e4 + 1.20e4i)T + (4.43e8 + 7.68e8i)T^{2} \) |
| 37 | \( 1 + (-3.74e4 - 6.48e4i)T + (-1.28e9 + 2.22e9i)T^{2} \) |
| 41 | \( 1 + 1.30e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 + 9.10e4T + 6.32e9T^{2} \) |
| 47 | \( 1 + (4.16e4 - 2.40e4i)T + (5.38e9 - 9.33e9i)T^{2} \) |
| 53 | \( 1 + (4.80e4 - 8.32e4i)T + (-1.10e10 - 1.91e10i)T^{2} \) |
| 59 | \( 1 + (-1.42e5 - 8.24e4i)T + (2.10e10 + 3.65e10i)T^{2} \) |
| 61 | \( 1 + (-1.07e5 + 6.18e4i)T + (2.57e10 - 4.46e10i)T^{2} \) |
| 67 | \( 1 + (2.37e5 - 4.10e5i)T + (-4.52e10 - 7.83e10i)T^{2} \) |
| 71 | \( 1 - 4.44e5T + 1.28e11T^{2} \) |
| 73 | \( 1 + (-4.50e5 - 2.60e5i)T + (7.56e10 + 1.31e11i)T^{2} \) |
| 79 | \( 1 + (9.70e4 + 1.68e5i)T + (-1.21e11 + 2.10e11i)T^{2} \) |
| 83 | \( 1 + 1.31e3iT - 3.26e11T^{2} \) |
| 89 | \( 1 + (1.96e5 - 1.13e5i)T + (2.48e11 - 4.30e11i)T^{2} \) |
| 97 | \( 1 + 4.70e5iT - 8.32e11T^{2} \) |
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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
−14.74254701483261970221706749865, −12.69338037104057298051262100689, −11.71827273438220607997055899024, −11.15502069381180979427834484059, −10.21685212449913466118606731938, −7.972554697502068079463938704783, −6.79783814678766457062568431506, −5.61648191739114222315905892379, −3.27761826627561727562286424894, −0.968072328599264831642434536656,
0.26403027857341565257887844818, 3.85368737976034441716735069954, 5.03759645575056309188731267851, 6.79013923741720417728226837543, 8.017842815624633022819399318492, 9.265881709903614518101850146290, 11.13012720588559670948190091054, 11.84771245033083375652258521741, 12.54855227787921063976510793691, 14.84545055943486013938719952099
