| L(s) = 1 | + (−0.340 − 1.69i)3-s + (1.68 − 0.296i)5-s + (0.981 − 1.16i)7-s + (−2.76 + 1.15i)9-s + (0.602 − 3.41i)11-s + (2.74 + 0.999i)13-s + (−1.07 − 2.75i)15-s + (0.812 − 0.469i)17-s + (−1.51 − 0.875i)19-s + (−2.32 − 1.26i)21-s + (−0.0294 + 0.0247i)23-s + (−1.95 + 0.712i)25-s + (2.90 + 4.30i)27-s + (−1.84 − 5.07i)29-s + (−4.26 − 5.08i)31-s + ⋯ |
| L(s) = 1 | + (−0.196 − 0.980i)3-s + (0.752 − 0.132i)5-s + (0.370 − 0.442i)7-s + (−0.922 + 0.385i)9-s + (0.181 − 1.03i)11-s + (0.761 + 0.277i)13-s + (−0.277 − 0.711i)15-s + (0.197 − 0.113i)17-s + (−0.348 − 0.200i)19-s + (−0.506 − 0.276i)21-s + (−0.00614 + 0.00515i)23-s + (−0.391 + 0.142i)25-s + (0.559 + 0.828i)27-s + (−0.343 − 0.942i)29-s + (−0.766 − 0.913i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0117 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0117 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.04369 - 1.03146i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.04369 - 1.03146i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.340 + 1.69i)T \) |
| good | 5 | \( 1 + (-1.68 + 0.296i)T + (4.69 - 1.71i)T^{2} \) |
| 7 | \( 1 + (-0.981 + 1.16i)T + (-1.21 - 6.89i)T^{2} \) |
| 11 | \( 1 + (-0.602 + 3.41i)T + (-10.3 - 3.76i)T^{2} \) |
| 13 | \( 1 + (-2.74 - 0.999i)T + (9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (-0.812 + 0.469i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.51 + 0.875i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.0294 - 0.0247i)T + (3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (1.84 + 5.07i)T + (-22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (4.26 + 5.08i)T + (-5.38 + 30.5i)T^{2} \) |
| 37 | \( 1 + (-3.09 - 5.36i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.60 + 4.41i)T + (-31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (-8.03 - 1.41i)T + (40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (6.29 + 5.27i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + 12.6iT - 53T^{2} \) |
| 59 | \( 1 + (-2.57 - 14.6i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (-10.6 - 8.91i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (4.52 - 12.4i)T + (-51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (-1.81 - 3.14i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (2.02 - 3.50i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-4.29 - 11.7i)T + (-60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (3.42 - 1.24i)T + (63.5 - 53.3i)T^{2} \) |
| 89 | \( 1 + (-7.61 - 4.39i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-0.737 + 4.18i)T + (-91.1 - 33.1i)T^{2} \) |
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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
−11.26018074425750940789255867738, −10.08858413309549766923518057253, −8.940784173510955057141961238295, −8.172099965924208625319048040693, −7.17604460902104819468988532518, −6.10520711041061105382375121223, −5.56013985346121692090211863824, −3.94009799546565216515258258474, −2.35243379656360758566055649075, −1.04226172225695994447559582188,
1.96592563997649857862948602838, 3.46462158698988188675575966935, 4.66993994718515951238018686146, 5.59507304460713143461182436627, 6.41996233102795307131463508108, 7.83487471846865612576429592335, 8.979280390994537421634090480265, 9.552713683626075188486441734937, 10.50683879584220664449616937335, 11.10196711257154594764926864187
