| L(s) = 1 | + (19.7 − 5.28i)2-s + (−11.4 + 42.8i)3-s + (250. − 144. i)4-s + (43.0 − 276. i)5-s + 906. i·6-s + (764. + 488. i)7-s + (2.32e3 − 2.32e3i)8-s + (187. + 108. i)9-s + (−609. − 5.67e3i)10-s + (−1.73e3 − 3.00e3i)11-s + (3.32e3 + 1.23e4i)12-s + (5.98e3 + 5.98e3i)13-s + (1.76e4 + 5.60e3i)14-s + (1.13e4 + 5.02e3i)15-s + (1.50e4 − 2.61e4i)16-s + (−1.44e4 − 3.86e3i)17-s + ⋯ |
| L(s) = 1 | + (1.74 − 0.467i)2-s + (−0.245 + 0.916i)3-s + (1.95 − 1.12i)4-s + (0.154 − 0.988i)5-s + 1.71i·6-s + (0.842 + 0.538i)7-s + (1.60 − 1.60i)8-s + (0.0857 + 0.0495i)9-s + (−0.192 − 1.79i)10-s + (−0.392 − 0.679i)11-s + (0.554 + 2.07i)12-s + (0.755 + 0.755i)13-s + (1.72 + 0.545i)14-s + (0.868 + 0.384i)15-s + (0.920 − 1.59i)16-s + (−0.711 − 0.190i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.956 + 0.292i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.956 + 0.292i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(4.65361 - 0.695711i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.65361 - 0.695711i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-43.0 + 276. i)T \) |
| 7 | \( 1 + (-764. - 488. i)T \) |
| good | 2 | \( 1 + (-19.7 + 5.28i)T + (110. - 64i)T^{2} \) |
| 3 | \( 1 + (11.4 - 42.8i)T + (-1.89e3 - 1.09e3i)T^{2} \) |
| 11 | \( 1 + (1.73e3 + 3.00e3i)T + (-9.74e6 + 1.68e7i)T^{2} \) |
| 13 | \( 1 + (-5.98e3 - 5.98e3i)T + 6.27e7iT^{2} \) |
| 17 | \( 1 + (1.44e4 + 3.86e3i)T + (3.55e8 + 2.05e8i)T^{2} \) |
| 19 | \( 1 + (1.61e4 - 2.80e4i)T + (-4.46e8 - 7.74e8i)T^{2} \) |
| 23 | \( 1 + (2.65e4 + 9.92e4i)T + (-2.94e9 + 1.70e9i)T^{2} \) |
| 29 | \( 1 + 4.78e4iT - 1.72e10T^{2} \) |
| 31 | \( 1 + (1.57e5 - 9.07e4i)T + (1.37e10 - 2.38e10i)T^{2} \) |
| 37 | \( 1 + (9.23e4 - 2.47e4i)T + (8.22e10 - 4.74e10i)T^{2} \) |
| 41 | \( 1 - 3.34e5iT - 1.94e11T^{2} \) |
| 43 | \( 1 + (6.72e4 - 6.72e4i)T - 2.71e11iT^{2} \) |
| 47 | \( 1 + (3.83e4 + 1.43e5i)T + (-4.38e11 + 2.53e11i)T^{2} \) |
| 53 | \( 1 + (-2.04e6 - 5.47e5i)T + (1.01e12 + 5.87e11i)T^{2} \) |
| 59 | \( 1 + (1.21e6 + 2.10e6i)T + (-1.24e12 + 2.15e12i)T^{2} \) |
| 61 | \( 1 + (-1.35e6 - 7.80e5i)T + (1.57e12 + 2.72e12i)T^{2} \) |
| 67 | \( 1 + (-1.37e5 + 5.12e5i)T + (-5.24e12 - 3.03e12i)T^{2} \) |
| 71 | \( 1 + 3.32e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + (-5.22e5 + 1.95e6i)T + (-9.56e12 - 5.52e12i)T^{2} \) |
| 79 | \( 1 + (-6.39e6 - 3.69e6i)T + (9.60e12 + 1.66e13i)T^{2} \) |
| 83 | \( 1 + (4.78e6 + 4.78e6i)T + 2.71e13iT^{2} \) |
| 89 | \( 1 + (-2.24e6 + 3.88e6i)T + (-2.21e13 - 3.83e13i)T^{2} \) |
| 97 | \( 1 + (-4.46e6 + 4.46e6i)T - 8.07e13iT^{2} \) |
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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
−14.78185603836567053950266911636, −13.68528984647408906902914023099, −12.59227732836223086831750636166, −11.47516384272825946368583385827, −10.51424917834977780282361326878, −8.643636462489757310962968635854, −6.00517077128828737809866560868, −4.89562906311766066993108599382, −4.06830339964356176645423019900, −1.90481526948055091927936633706,
2.06478319222347228633947641363, 3.89279766884402610909466681229, 5.59262191985083727048440283165, 6.89662966951063711958027852349, 7.58627402094457134424599196746, 10.75577970693500235251180517339, 11.73833043179067954650816126643, 13.12866673878450623640528458079, 13.61318029211107742731228532589, 14.93527030863641541344528818112
