| L(s) = 1 | + (−5.59 − 0.843i)2-s + (23.8 + 34.9i)3-s + (30.5 + 9.43i)4-s + (32.1 − 2.40i)5-s + (−103. − 215. i)6-s + (−129. − 317. i)7-s + (−163. − 78.5i)8-s + (−388. + 990. i)9-s + (−181. − 13.6i)10-s + (580. + 1.47e3i)11-s + (399. + 1.29e3i)12-s + (1.31e3 + 1.04e3i)13-s + (456. + 1.88e3i)14-s + (850. + 1.06e3i)15-s + (846. + 576. i)16-s + (−4.93e3 + 5.31e3i)17-s + ⋯ |
| L(s) = 1 | + (−0.699 − 0.105i)2-s + (0.883 + 1.29i)3-s + (0.477 + 0.147i)4-s + (0.257 − 0.0192i)5-s + (−0.481 − 0.999i)6-s + (−0.377 − 0.926i)7-s + (−0.318 − 0.153i)8-s + (−0.533 + 1.35i)9-s + (−0.181 − 0.0136i)10-s + (0.436 + 1.11i)11-s + (0.231 + 0.749i)12-s + (0.596 + 0.476i)13-s + (0.166 + 0.687i)14-s + (0.252 + 0.316i)15-s + (0.206 + 0.140i)16-s + (−1.00 + 1.08i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.713 - 0.700i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.713 - 0.700i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.575616 + 1.40886i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.575616 + 1.40886i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (5.59 + 0.843i)T \) |
| 7 | \( 1 + (129. + 317. i)T \) |
| good | 3 | \( 1 + (-23.8 - 34.9i)T + (-266. + 678. i)T^{2} \) |
| 5 | \( 1 + (-32.1 + 2.40i)T + (1.54e4 - 2.32e3i)T^{2} \) |
| 11 | \( 1 + (-580. - 1.47e3i)T + (-1.29e6 + 1.20e6i)T^{2} \) |
| 13 | \( 1 + (-1.31e3 - 1.04e3i)T + (1.07e6 + 4.70e6i)T^{2} \) |
| 17 | \( 1 + (4.93e3 - 5.31e3i)T + (-1.80e6 - 2.40e7i)T^{2} \) |
| 19 | \( 1 + (-5.44e3 + 3.14e3i)T + (2.35e7 - 4.07e7i)T^{2} \) |
| 23 | \( 1 + (1.47e4 - 1.36e4i)T + (1.10e7 - 1.47e8i)T^{2} \) |
| 29 | \( 1 + (-3.74e3 + 1.63e4i)T + (-5.35e8 - 2.58e8i)T^{2} \) |
| 31 | \( 1 + (-4.51e4 - 2.60e4i)T + (4.43e8 + 7.68e8i)T^{2} \) |
| 37 | \( 1 + (6.09e4 - 1.88e4i)T + (2.11e9 - 1.44e9i)T^{2} \) |
| 41 | \( 1 + (-2.55e4 + 5.30e4i)T + (-2.96e9 - 3.71e9i)T^{2} \) |
| 43 | \( 1 + (7.08e4 - 3.41e4i)T + (3.94e9 - 4.94e9i)T^{2} \) |
| 47 | \( 1 + (1.04e4 - 6.96e4i)T + (-1.03e10 - 3.17e9i)T^{2} \) |
| 53 | \( 1 + (3.87e4 + 1.19e4i)T + (1.83e10 + 1.24e10i)T^{2} \) |
| 59 | \( 1 + (1.27e5 + 9.52e3i)T + (4.17e10 + 6.28e9i)T^{2} \) |
| 61 | \( 1 + (-3.91e4 - 1.26e5i)T + (-4.25e10 + 2.90e10i)T^{2} \) |
| 67 | \( 1 + (-2.30e5 + 3.98e5i)T + (-4.52e10 - 7.83e10i)T^{2} \) |
| 71 | \( 1 + (-1.23e5 - 5.41e5i)T + (-1.15e11 + 5.55e10i)T^{2} \) |
| 73 | \( 1 + (4.95e4 + 3.28e5i)T + (-1.44e11 + 4.46e10i)T^{2} \) |
| 79 | \( 1 + (3.49e5 + 6.05e5i)T + (-1.21e11 + 2.10e11i)T^{2} \) |
| 83 | \( 1 + (-3.01e5 + 2.40e5i)T + (7.27e10 - 3.18e11i)T^{2} \) |
| 89 | \( 1 + (-2.60e5 - 1.02e5i)T + (3.64e11 + 3.38e11i)T^{2} \) |
| 97 | \( 1 - 4.43e4iT - 8.32e11T^{2} \) |
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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
−13.44378466548089751599590862372, −11.76733398649878734583759248194, −10.45688540999990623115242317541, −9.849047362947769633874091158949, −9.083476010860603913573859740299, −7.899865821056007505604962416693, −6.52015495330879779517078467896, −4.43791150518978419379471993141, −3.51830065392495344613046568934, −1.78326775187718164720674301940,
0.57390978128664771119236407953, 2.03707962145928130627925595364, 3.10447854919255468555147847430, 5.91467427776323174985457045105, 6.74571376400061178752736814851, 8.174549010422545898936041855733, 8.668908229495571065675454596085, 9.806241692473554194719438562618, 11.46942008850770349685792468115, 12.33114931419194495042971850559
