| L(s) = 1 | + (−0.654 − 1.25i)2-s + (−1.14 + 1.64i)4-s + (−0.137 − 0.0242i)5-s + (−2.98 − 3.56i)7-s + (2.80 + 0.361i)8-s + (0.0594 + 0.187i)10-s + (0.182 + 1.03i)11-s + (−1.46 + 0.532i)13-s + (−2.51 + 6.07i)14-s + (−1.38 − 3.75i)16-s + (−5.25 − 3.03i)17-s + (−3.80 + 2.19i)19-s + (0.196 − 0.197i)20-s + (1.17 − 0.906i)22-s + (−1.94 − 1.62i)23-s + ⋯ |
| L(s) = 1 | + (−0.462 − 0.886i)2-s + (−0.572 + 0.820i)4-s + (−0.0613 − 0.0108i)5-s + (−1.12 − 1.34i)7-s + (0.991 + 0.127i)8-s + (0.0187 + 0.0594i)10-s + (0.0550 + 0.312i)11-s + (−0.405 + 0.147i)13-s + (−0.670 + 1.62i)14-s + (−0.345 − 0.938i)16-s + (−1.27 − 0.736i)17-s + (−0.871 + 0.503i)19-s + (0.0439 − 0.0441i)20-s + (0.251 − 0.193i)22-s + (−0.404 − 0.339i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.955 - 0.294i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.955 - 0.294i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0530861 + 0.352116i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0530861 + 0.352116i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.654 + 1.25i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (0.137 + 0.0242i)T + (4.69 + 1.71i)T^{2} \) |
| 7 | \( 1 + (2.98 + 3.56i)T + (-1.21 + 6.89i)T^{2} \) |
| 11 | \( 1 + (-0.182 - 1.03i)T + (-10.3 + 3.76i)T^{2} \) |
| 13 | \( 1 + (1.46 - 0.532i)T + (9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (5.25 + 3.03i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (3.80 - 2.19i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.94 + 1.62i)T + (3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (0.210 - 0.578i)T + (-22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (-3.18 + 3.79i)T + (-5.38 - 30.5i)T^{2} \) |
| 37 | \( 1 + (1.43 - 2.49i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-2.25 - 6.18i)T + (-31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (-2.15 + 0.380i)T + (40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (-8.46 + 7.10i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + 8.73iT - 53T^{2} \) |
| 59 | \( 1 + (-1.42 + 8.08i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (-8.56 + 7.18i)T + (10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (-1.93 - 5.30i)T + (-51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (2.33 - 4.04i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (5.64 + 9.77i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.572 + 1.57i)T + (-60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (2.20 + 0.802i)T + (63.5 + 53.3i)T^{2} \) |
| 89 | \( 1 + (7.63 - 4.40i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (0.978 + 5.54i)T + (-91.1 + 33.1i)T^{2} \) |
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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
−10.98841607328627431905833768700, −10.06673521607670758899792505717, −9.635912795046182412170250067918, −8.421193035989902047824215597869, −7.33227528247204661105795600698, −6.49960884945848400997482699484, −4.51554000736518404889507427228, −3.76755052655474992026553719419, −2.31895241814242645103348809422, −0.27824460836362598986058886450,
2.39231860697807280070475747477, 4.17265980868493140351699743484, 5.65674628016749834471919726491, 6.23278536380278435458892905609, 7.25919105249332932357773061423, 8.610100809917781695520735642702, 9.026241392538560734757335920989, 9.998095907163611866128281545768, 11.00100873794668253200402270183, 12.26939598488131992009015036136
