| 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
−14.84545055943486013938719952099, −12.54855227787921063976510793691, −11.84771245033083375652258521741, −11.13012720588559670948190091054, −9.265881709903614518101850146290, −8.017842815624633022819399318492, −6.79013923741720417728226837543, −5.03759645575056309188731267851, −3.85368737976034441716735069954, −0.26403027857341565257887844818,
0.968072328599264831642434536656, 3.27761826627561727562286424894, 5.61648191739114222315905892379, 6.79783814678766457062568431506, 7.972554697502068079463938704783, 10.21685212449913466118606731938, 11.15502069381180979427834484059, 11.71827273438220607997055899024, 12.69338037104057298051262100689, 14.74254701483261970221706749865
