| L(s) = 1 | + (−1.87 + 0.684i)2-s + (3.06 − 2.57i)4-s + (−0.127 − 0.721i)5-s + (1.18 + 0.994i)7-s + (−4.00 + 6.92i)8-s + (0.733 + 1.26i)10-s + (4.18 − 23.7i)11-s + (26.9 + 9.79i)13-s + (−2.90 − 1.05i)14-s + (2.77 − 15.7i)16-s + (−34.8 − 60.3i)17-s + (−14.7 + 25.5i)19-s + (−2.24 − 1.88i)20-s + (8.36 + 47.4i)22-s + (125. − 105. i)23-s + ⋯ |
| L(s) = 1 | + (−0.664 + 0.241i)2-s + (0.383 − 0.321i)4-s + (−0.0113 − 0.0645i)5-s + (0.0640 + 0.0537i)7-s + (−0.176 + 0.306i)8-s + (0.0231 + 0.0401i)10-s + (0.114 − 0.649i)11-s + (0.573 + 0.208i)13-s + (−0.0555 − 0.0202i)14-s + (0.0434 − 0.246i)16-s + (−0.497 − 0.861i)17-s + (−0.178 + 0.308i)19-s + (−0.0251 − 0.0210i)20-s + (0.0810 + 0.459i)22-s + (1.13 − 0.956i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.854 + 0.519i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.854 + 0.519i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.15896 - 0.324550i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.15896 - 0.324550i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.87 - 0.684i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (0.127 + 0.721i)T + (-117. + 42.7i)T^{2} \) |
| 7 | \( 1 + (-1.18 - 0.994i)T + (59.5 + 337. i)T^{2} \) |
| 11 | \( 1 + (-4.18 + 23.7i)T + (-1.25e3 - 455. i)T^{2} \) |
| 13 | \( 1 + (-26.9 - 9.79i)T + (1.68e3 + 1.41e3i)T^{2} \) |
| 17 | \( 1 + (34.8 + 60.3i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (14.7 - 25.5i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-125. + 105. i)T + (2.11e3 - 1.19e4i)T^{2} \) |
| 29 | \( 1 + (-197. + 71.7i)T + (1.86e4 - 1.56e4i)T^{2} \) |
| 31 | \( 1 + (-83.3 + 69.9i)T + (5.17e3 - 2.93e4i)T^{2} \) |
| 37 | \( 1 + (-63.3 - 109. i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + (-95.0 - 34.5i)T + (5.27e4 + 4.43e4i)T^{2} \) |
| 43 | \( 1 + (-85.0 + 482. i)T + (-7.47e4 - 2.71e4i)T^{2} \) |
| 47 | \( 1 + (-223. - 187. i)T + (1.80e4 + 1.02e5i)T^{2} \) |
| 53 | \( 1 + 523.T + 1.48e5T^{2} \) |
| 59 | \( 1 + (-100. - 572. i)T + (-1.92e5 + 7.02e4i)T^{2} \) |
| 61 | \( 1 + (-104. - 87.7i)T + (3.94e4 + 2.23e5i)T^{2} \) |
| 67 | \( 1 + (822. + 299. i)T + (2.30e5 + 1.93e5i)T^{2} \) |
| 71 | \( 1 + (265. + 460. i)T + (-1.78e5 + 3.09e5i)T^{2} \) |
| 73 | \( 1 + (593. - 1.02e3i)T + (-1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (200. - 72.8i)T + (3.77e5 - 3.16e5i)T^{2} \) |
| 83 | \( 1 + (-506. + 184. i)T + (4.38e5 - 3.67e5i)T^{2} \) |
| 89 | \( 1 + (173. - 300. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + (83.7 - 475. i)T + (-8.57e5 - 3.12e5i)T^{2} \) |
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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
−12.13171501417571628993821188549, −11.14688220501602888266487037921, −10.30729008892504936141632231259, −9.006953548339887548997821897276, −8.409095866986817936338656582137, −7.04238079890004482922909400725, −6.08745145466786732299584914767, −4.61315330928775750770894526494, −2.76405276412345311429289314781, −0.822077714070758967710107350496,
1.32807808455159769810546520604, 3.04039376570014916850209521683, 4.63190784008056011512529848836, 6.29255902254311139360063669592, 7.34663469926633001100237745880, 8.515968994391805426195595269835, 9.383146250118010076635847399683, 10.56758176844860200396951708818, 11.21948511274489045587129926513, 12.46367244531016382692090146071
