| L(s) = 1 | + (−0.929 − 1.06i)2-s + (1.69 + 0.353i)3-s + (−0.270 + 1.98i)4-s + (2.09 + 2.39i)5-s + (−1.20 − 2.13i)6-s + (1.56 + 0.205i)7-s + (2.36 − 1.55i)8-s + (2.75 + 1.19i)9-s + (0.597 − 4.45i)10-s + (−1.23 − 2.50i)11-s + (−1.15 + 3.26i)12-s + (−1.88 − 5.54i)13-s + (−1.23 − 1.85i)14-s + (2.71 + 4.79i)15-s + (−3.85 − 1.07i)16-s + (1.62 + 1.62i)17-s + ⋯ |
| L(s) = 1 | + (−0.657 − 0.753i)2-s + (0.978 + 0.203i)3-s + (−0.135 + 0.990i)4-s + (0.937 + 1.06i)5-s + (−0.490 − 0.871i)6-s + (0.591 + 0.0778i)7-s + (0.835 − 0.549i)8-s + (0.916 + 0.399i)9-s + (0.188 − 1.40i)10-s + (−0.372 − 0.755i)11-s + (−0.334 + 0.942i)12-s + (−0.522 − 1.53i)13-s + (−0.330 − 0.496i)14-s + (0.700 + 1.23i)15-s + (−0.963 − 0.267i)16-s + (0.394 + 0.394i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0429i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0429i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.77741 - 0.0382209i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.77741 - 0.0382209i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.929 + 1.06i)T \) |
| 3 | \( 1 + (-1.69 - 0.353i)T \) |
| good | 5 | \( 1 + (-2.09 - 2.39i)T + (-0.652 + 4.95i)T^{2} \) |
| 7 | \( 1 + (-1.56 - 0.205i)T + (6.76 + 1.81i)T^{2} \) |
| 11 | \( 1 + (1.23 + 2.50i)T + (-6.69 + 8.72i)T^{2} \) |
| 13 | \( 1 + (1.88 + 5.54i)T + (-10.3 + 7.91i)T^{2} \) |
| 17 | \( 1 + (-1.62 - 1.62i)T + 17iT^{2} \) |
| 19 | \( 1 + (0.537 - 2.70i)T + (-17.5 - 7.27i)T^{2} \) |
| 23 | \( 1 + (-0.443 - 3.36i)T + (-22.2 + 5.95i)T^{2} \) |
| 29 | \( 1 + (-9.16 + 0.600i)T + (28.7 - 3.78i)T^{2} \) |
| 31 | \( 1 + (2.25 - 1.30i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (1.63 + 8.22i)T + (-34.1 + 14.1i)T^{2} \) |
| 41 | \( 1 + (-0.727 - 5.52i)T + (-39.6 + 10.6i)T^{2} \) |
| 43 | \( 1 + (5.29 - 2.61i)T + (26.1 - 34.1i)T^{2} \) |
| 47 | \( 1 + (7.82 - 2.09i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (7.10 - 10.6i)T + (-20.2 - 48.9i)T^{2} \) |
| 59 | \( 1 + (1.06 + 1.21i)T + (-7.70 + 58.4i)T^{2} \) |
| 61 | \( 1 + (0.437 + 6.67i)T + (-60.4 + 7.96i)T^{2} \) |
| 67 | \( 1 + (-3.49 - 1.72i)T + (40.7 + 53.1i)T^{2} \) |
| 71 | \( 1 + (6.37 + 15.3i)T + (-50.2 + 50.2i)T^{2} \) |
| 73 | \( 1 + (-1.67 + 4.04i)T + (-51.6 - 51.6i)T^{2} \) |
| 79 | \( 1 + (1.87 + 6.98i)T + (-68.4 + 39.5i)T^{2} \) |
| 83 | \( 1 + (0.218 + 0.191i)T + (10.8 + 82.2i)T^{2} \) |
| 89 | \( 1 + (4.02 - 1.66i)T + (62.9 - 62.9i)T^{2} \) |
| 97 | \( 1 + (-7.36 - 4.25i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.37644590010246038578533931542, −10.11522937972952133698164457759, −9.111974236933089673480581884025, −8.005002333330152744038089810030, −7.73466434953460913864880502248, −6.31204733785418543861489238067, −4.97768455420092284719603055188, −3.33393650152052366432589208811, −2.84909263709652774792395095060, −1.66231023596388462190867216583,
1.40370796694320263305173015958, 2.26982180423064492950011443170, 4.62215802865868966969646825742, 4.96184948062279720017015612046, 6.54424860669952348325031658870, 7.19685914675975532444763873233, 8.374855124571679248973287520190, 8.765759858952885794601779976744, 9.748296996050517159183389752355, 10.02731670197199988565100688777
