| L(s) = 1 | + (−0.326 − 1.37i)2-s + (0.243 − 2.99i)3-s + (−1.78 + 0.897i)4-s + (−2.26 + 1.68i)5-s + (−4.19 + 0.640i)6-s + (−8.30 + 5.46i)7-s + (1.81 + 2.16i)8-s + (−8.88 − 1.45i)9-s + (3.06 + 2.57i)10-s + (−1.30 + 11.1i)11-s + (2.24 + 5.56i)12-s + (−4.43 − 14.8i)13-s + (10.2 + 9.64i)14-s + (4.49 + 7.19i)15-s + (2.38 − 3.20i)16-s + (−4.36 − 0.768i)17-s + ⋯ |
| L(s) = 1 | + (−0.163 − 0.688i)2-s + (0.0810 − 0.996i)3-s + (−0.446 + 0.224i)4-s + (−0.453 + 0.337i)5-s + (−0.698 + 0.106i)6-s + (−1.18 + 0.780i)7-s + (0.227 + 0.270i)8-s + (−0.986 − 0.161i)9-s + (0.306 + 0.257i)10-s + (−0.118 + 1.01i)11-s + (0.187 + 0.463i)12-s + (−0.340 − 1.13i)13-s + (0.730 + 0.689i)14-s + (0.299 + 0.479i)15-s + (0.149 − 0.200i)16-s + (−0.256 − 0.0452i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.383 - 0.923i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.383 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0214143 + 0.0320626i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0214143 + 0.0320626i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.326 + 1.37i)T \) |
| 3 | \( 1 + (-0.243 + 2.99i)T \) |
| good | 5 | \( 1 + (2.26 - 1.68i)T + (7.17 - 23.9i)T^{2} \) |
| 7 | \( 1 + (8.30 - 5.46i)T + (19.4 - 44.9i)T^{2} \) |
| 11 | \( 1 + (1.30 - 11.1i)T + (-117. - 27.9i)T^{2} \) |
| 13 | \( 1 + (4.43 + 14.8i)T + (-141. + 92.8i)T^{2} \) |
| 17 | \( 1 + (4.36 + 0.768i)T + (271. + 98.8i)T^{2} \) |
| 19 | \( 1 + (1.93 + 10.9i)T + (-339. + 123. i)T^{2} \) |
| 23 | \( 1 + (-3.87 + 5.89i)T + (-209. - 485. i)T^{2} \) |
| 29 | \( 1 + (27.5 - 25.9i)T + (48.8 - 839. i)T^{2} \) |
| 31 | \( 1 + (-3.19 + 54.8i)T + (-954. - 111. i)T^{2} \) |
| 37 | \( 1 + (37.2 - 13.5i)T + (1.04e3 - 879. i)T^{2} \) |
| 41 | \( 1 + (9.88 - 41.7i)T + (-1.50e3 - 754. i)T^{2} \) |
| 43 | \( 1 + (27.3 + 63.3i)T + (-1.26e3 + 1.34e3i)T^{2} \) |
| 47 | \( 1 + (-62.3 + 3.63i)T + (2.19e3 - 256. i)T^{2} \) |
| 53 | \( 1 + (63.7 + 36.8i)T + (1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (2.12 + 18.2i)T + (-3.38e3 + 802. i)T^{2} \) |
| 61 | \( 1 + (-35.5 - 17.8i)T + (2.22e3 + 2.98e3i)T^{2} \) |
| 67 | \( 1 + (-12.7 + 13.5i)T + (-261. - 4.48e3i)T^{2} \) |
| 71 | \( 1 + (16.0 - 19.1i)T + (-875. - 4.96e3i)T^{2} \) |
| 73 | \( 1 + (-46.5 + 39.0i)T + (925. - 5.24e3i)T^{2} \) |
| 79 | \( 1 + (51.7 - 12.2i)T + (5.57e3 - 2.80e3i)T^{2} \) |
| 83 | \( 1 + (-2.26 - 9.56i)T + (-6.15e3 + 3.09e3i)T^{2} \) |
| 89 | \( 1 + (3.97 + 4.73i)T + (-1.37e3 + 7.80e3i)T^{2} \) |
| 97 | \( 1 + (31.9 - 42.9i)T + (-2.69e3 - 9.01e3i)T^{2} \) |
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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
−12.14729844800798346284639789659, −11.13888200538273992870406265880, −9.894900274157616658960612595060, −8.945935140327776066962488656712, −7.71366153131002856648945968175, −6.78314547806720777485844633504, −5.40295218370081637601306091058, −3.36424183337374136220323042655, −2.30727322060912197208154908130, −0.02366581238221795714640219730,
3.46800988946220361704835895745, 4.40823053824415920592947522267, 5.84082828158842462260412941299, 6.97001562552043224949329505332, 8.330237394123974481057717180208, 9.219950600920720002796191540245, 10.09368947930182956888369673225, 11.08120245585768160922291708847, 12.35073761434481943229269506132, 13.69275792006127798197557783541
