| L(s) = 1 | + (−4.87 − 1.91i)2-s + (−5.19 − 0.0355i)3-s + (14.2 + 13.1i)4-s + (3.32 + 2.26i)5-s + (25.2 + 10.1i)6-s + (1.07 + 18.4i)7-s + (−25.8 − 53.6i)8-s + (26.9 + 0.369i)9-s + (−11.8 − 17.3i)10-s + (−6.81 − 45.2i)11-s + (−73.3 − 69.0i)12-s + (−16.7 + 13.3i)13-s + (30.0 − 92.1i)14-s + (−17.1 − 11.8i)15-s + (11.7 + 156. i)16-s + (13.5 + 4.18i)17-s + ⋯ |
| L(s) = 1 | + (−1.72 − 0.675i)2-s + (−0.999 − 0.00684i)3-s + (1.77 + 1.64i)4-s + (0.297 + 0.202i)5-s + (1.71 + 0.687i)6-s + (0.0582 + 0.998i)7-s + (−1.14 − 2.37i)8-s + (0.999 + 0.0136i)9-s + (−0.374 − 0.549i)10-s + (−0.186 − 1.23i)11-s + (−1.76 − 1.66i)12-s + (−0.356 + 0.284i)13-s + (0.574 − 1.75i)14-s + (−0.295 − 0.204i)15-s + (0.183 + 2.44i)16-s + (0.193 + 0.0596i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.377 - 0.925i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.377 - 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.124465 + 0.185252i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.124465 + 0.185252i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (5.19 + 0.0355i)T \) |
| 7 | \( 1 + (-1.07 - 18.4i)T \) |
| good | 2 | \( 1 + (4.87 + 1.91i)T + (5.86 + 5.44i)T^{2} \) |
| 5 | \( 1 + (-3.32 - 2.26i)T + (45.6 + 116. i)T^{2} \) |
| 11 | \( 1 + (6.81 + 45.2i)T + (-1.27e3 + 392. i)T^{2} \) |
| 13 | \( 1 + (16.7 - 13.3i)T + (488. - 2.14e3i)T^{2} \) |
| 17 | \( 1 + (-13.5 - 4.18i)T + (4.05e3 + 2.76e3i)T^{2} \) |
| 19 | \( 1 + (-48.0 + 27.7i)T + (3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-48.9 - 158. i)T + (-1.00e4 + 6.85e3i)T^{2} \) |
| 29 | \( 1 + (220. - 50.2i)T + (2.19e4 - 1.05e4i)T^{2} \) |
| 31 | \( 1 + (-262. - 151. i)T + (1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (106. - 98.3i)T + (3.78e3 - 5.05e4i)T^{2} \) |
| 41 | \( 1 + (177. - 85.3i)T + (4.29e4 - 5.38e4i)T^{2} \) |
| 43 | \( 1 + (93.3 + 44.9i)T + (4.95e4 + 6.21e4i)T^{2} \) |
| 47 | \( 1 + (-90.1 + 229. i)T + (-7.61e4 - 7.06e4i)T^{2} \) |
| 53 | \( 1 + (345. - 372. i)T + (-1.11e4 - 1.48e5i)T^{2} \) |
| 59 | \( 1 + (343. - 234. i)T + (7.50e4 - 1.91e5i)T^{2} \) |
| 61 | \( 1 + (393. + 424. i)T + (-1.69e4 + 2.26e5i)T^{2} \) |
| 67 | \( 1 + (293. - 509. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + (222. + 50.8i)T + (3.22e5 + 1.55e5i)T^{2} \) |
| 73 | \( 1 + (474. - 186. i)T + (2.85e5 - 2.64e5i)T^{2} \) |
| 79 | \( 1 + (-349. - 605. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + (790. - 991. i)T + (-1.27e5 - 5.57e5i)T^{2} \) |
| 89 | \( 1 + (-806. - 121. i)T + (6.73e5 + 2.07e5i)T^{2} \) |
| 97 | \( 1 - 347. iT - 9.12e5T^{2} \) |
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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
−12.17571803852763998135742473613, −11.62941861836677148386383586425, −10.84213053720382148197848229539, −9.863888611468971138342471939116, −9.029701513097733884739791829859, −7.899976590625552577138538629360, −6.67755846138583932698079167735, −5.49557308945476103111042162566, −3.03202010333236773417580471809, −1.43436034460050357216302617037,
0.22991753039299104188190529248, 1.61851675582611691711231938793, 4.79511217639753330139135816719, 6.12293262147068556288941766849, 7.19157127312648925076499085337, 7.76945305890729796773526095087, 9.426925088841423009414307508502, 10.08393136340616417818231876617, 10.76353026434798176928042148708, 11.89136253887341538365800713657
