| L(s) = 1 | + (−7.93 + 0.834i)2-s + (−69.1 + 50.2i)3-s + (−62.9 + 13.3i)4-s + (−215. − 239. i)5-s + (507. − 456. i)6-s + (51.7 + 116. i)7-s + (1.45e3 − 474. i)8-s + (1.58e3 − 4.87e3i)9-s + (1.90e3 + 1.71e3i)10-s − 4.54e3i·11-s + (3.68e3 − 4.08e3i)12-s + (−5.69e3 + 9.86e3i)13-s + (−507. − 879. i)14-s + (2.69e4 + 5.72e3i)15-s + (−3.66e3 + 1.63e3i)16-s + (−5.01e3 − 2.35e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.701 + 0.0737i)2-s + (−1.47 + 1.07i)3-s + (−0.491 + 0.104i)4-s + (−0.770 − 0.856i)5-s + (0.958 − 0.863i)6-s + (0.0570 + 0.128i)7-s + (1.00 − 0.327i)8-s + (0.724 − 2.23i)9-s + (0.603 + 0.543i)10-s − 1.02i·11-s + (0.615 − 0.683i)12-s + (−0.719 + 1.24i)13-s + (−0.0494 − 0.0856i)14-s + (2.06 + 0.438i)15-s + (−0.223 + 0.0995i)16-s + (−0.247 − 1.16i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.243 - 0.969i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.243 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.0505715 + 0.0648482i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0505715 + 0.0648482i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 61 | \( 1 + (1.40e6 + 1.08e6i)T \) |
| good | 2 | \( 1 + (7.93 - 0.834i)T + (125. - 26.6i)T^{2} \) |
| 3 | \( 1 + (69.1 - 50.2i)T + (675. - 2.07e3i)T^{2} \) |
| 5 | \( 1 + (215. + 239. i)T + (-8.16e3 + 7.76e4i)T^{2} \) |
| 7 | \( 1 + (-51.7 - 116. i)T + (-5.51e5 + 6.12e5i)T^{2} \) |
| 11 | \( 1 + 4.54e3iT - 1.94e7T^{2} \) |
| 13 | \( 1 + (5.69e3 - 9.86e3i)T + (-3.13e7 - 5.43e7i)T^{2} \) |
| 17 | \( 1 + (5.01e3 + 2.35e4i)T + (-3.74e8 + 1.66e8i)T^{2} \) |
| 19 | \( 1 + (3.85e4 + 1.71e4i)T + (5.98e8 + 6.64e8i)T^{2} \) |
| 23 | \( 1 + (1.00e5 + 3.24e4i)T + (2.75e9 + 2.00e9i)T^{2} \) |
| 29 | \( 1 + (-2.05e5 + 1.18e5i)T + (8.62e9 - 1.49e10i)T^{2} \) |
| 31 | \( 1 + (-4.07e4 - 4.28e3i)T + (2.69e10 + 5.72e9i)T^{2} \) |
| 37 | \( 1 + (1.91e5 - 2.63e5i)T + (-2.93e10 - 9.02e10i)T^{2} \) |
| 41 | \( 1 + (-2.63e5 - 1.91e5i)T + (6.01e10 + 1.85e11i)T^{2} \) |
| 43 | \( 1 + (2.54e4 - 1.19e5i)T + (-2.48e11 - 1.10e11i)T^{2} \) |
| 47 | \( 1 + (2.73e5 + 4.74e5i)T + (-2.53e11 + 4.38e11i)T^{2} \) |
| 53 | \( 1 + (7.34e5 - 2.38e5i)T + (9.50e11 - 6.90e11i)T^{2} \) |
| 59 | \( 1 + (2.44e6 - 2.56e5i)T + (2.43e12 - 5.17e11i)T^{2} \) |
| 67 | \( 1 + (5.43e5 - 4.89e5i)T + (6.33e11 - 6.02e12i)T^{2} \) |
| 71 | \( 1 + (6.75e5 + 6.08e5i)T + (9.50e11 + 9.04e12i)T^{2} \) |
| 73 | \( 1 + (1.16e6 - 1.29e6i)T + (-1.15e12 - 1.09e13i)T^{2} \) |
| 79 | \( 1 + (1.48e5 - 6.98e5i)T + (-1.75e13 - 7.81e12i)T^{2} \) |
| 83 | \( 1 + (-2.10e5 - 1.99e6i)T + (-2.65e13 + 5.64e12i)T^{2} \) |
| 89 | \( 1 + (3.33e5 + 4.59e5i)T + (-1.36e13 + 4.20e13i)T^{2} \) |
| 97 | \( 1 + (1.15e6 - 1.10e7i)T + (-7.90e13 - 1.67e13i)T^{2} \) |
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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
−13.90227338287403263177059040034, −12.22861747174656330228378329359, −11.59340799249472495484292742520, −10.36542760679768266352316520557, −9.320315255664458314172405977121, −8.342161377325974745450756819630, −6.45253081083392353276103871620, −4.71608591579393388428610815186, −4.32443734062168688528409294813, −0.53277835728861597205672808575,
0.10515572222730698479546715260, 1.73770044551070213140514803545, 4.48570716355189320058266184050, 6.02043032441108631697692992537, 7.35409241261822268181925573280, 8.031728402289349198200444916363, 10.38024455454559796016962914498, 10.65127200315823765083047360684, 12.19169486355912331885907256514, 12.76301731117290603054803508384
