| L(s) = 1 | + (−0.396 − 2.75i)3-s + (−1.50 − 0.441i)5-s + (−0.107 + 0.124i)7-s + (−4.57 + 1.34i)9-s + (5.36 − 3.44i)11-s + (2.99 + 3.45i)13-s + (−0.621 + 4.32i)15-s + (−0.232 + 0.510i)17-s + (1.75 + 3.85i)19-s + (0.386 + 0.248i)21-s + (−2.40 − 4.14i)23-s + (−2.13 − 1.37i)25-s + (2.04 + 4.47i)27-s + (−2.34 + 5.12i)29-s + (−1.27 + 8.86i)31-s + ⋯ |
| L(s) = 1 | + (−0.228 − 1.59i)3-s + (−0.672 − 0.197i)5-s + (−0.0407 + 0.0470i)7-s + (−1.52 + 0.447i)9-s + (1.61 − 1.03i)11-s + (0.829 + 0.957i)13-s + (−0.160 + 1.11i)15-s + (−0.0565 + 0.123i)17-s + (0.403 + 0.883i)19-s + (0.0843 + 0.0541i)21-s + (−0.501 − 0.865i)23-s + (−0.427 − 0.274i)25-s + (0.393 + 0.861i)27-s + (−0.434 + 0.951i)29-s + (−0.228 + 1.59i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00508 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.00508 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.617620 - 0.620767i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.617620 - 0.620767i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 23 | \( 1 + (2.40 + 4.14i)T \) |
| good | 3 | \( 1 + (0.396 + 2.75i)T + (-2.87 + 0.845i)T^{2} \) |
| 5 | \( 1 + (1.50 + 0.441i)T + (4.20 + 2.70i)T^{2} \) |
| 7 | \( 1 + (0.107 - 0.124i)T + (-0.996 - 6.92i)T^{2} \) |
| 11 | \( 1 + (-5.36 + 3.44i)T + (4.56 - 10.0i)T^{2} \) |
| 13 | \( 1 + (-2.99 - 3.45i)T + (-1.85 + 12.8i)T^{2} \) |
| 17 | \( 1 + (0.232 - 0.510i)T + (-11.1 - 12.8i)T^{2} \) |
| 19 | \( 1 + (-1.75 - 3.85i)T + (-12.4 + 14.3i)T^{2} \) |
| 29 | \( 1 + (2.34 - 5.12i)T + (-18.9 - 21.9i)T^{2} \) |
| 31 | \( 1 + (1.27 - 8.86i)T + (-29.7 - 8.73i)T^{2} \) |
| 37 | \( 1 + (-5.69 + 1.67i)T + (31.1 - 20.0i)T^{2} \) |
| 41 | \( 1 + (-3.37 - 0.991i)T + (34.4 + 22.1i)T^{2} \) |
| 43 | \( 1 + (1.17 + 8.15i)T + (-41.2 + 12.1i)T^{2} \) |
| 47 | \( 1 + 5.52T + 47T^{2} \) |
| 53 | \( 1 + (-1.22 + 1.41i)T + (-7.54 - 52.4i)T^{2} \) |
| 59 | \( 1 + (4.03 + 4.65i)T + (-8.39 + 58.3i)T^{2} \) |
| 61 | \( 1 + (0.202 - 1.40i)T + (-58.5 - 17.1i)T^{2} \) |
| 67 | \( 1 + (6.36 + 4.09i)T + (27.8 + 60.9i)T^{2} \) |
| 71 | \( 1 + (1.68 + 1.08i)T + (29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (-4.50 - 9.85i)T + (-47.8 + 55.1i)T^{2} \) |
| 79 | \( 1 + (-5.70 - 6.58i)T + (-11.2 + 78.1i)T^{2} \) |
| 83 | \( 1 + (-0.778 + 0.228i)T + (69.8 - 44.8i)T^{2} \) |
| 89 | \( 1 + (-0.923 - 6.42i)T + (-85.3 + 25.0i)T^{2} \) |
| 97 | \( 1 + (7.25 + 2.13i)T + (81.6 + 52.4i)T^{2} \) |
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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
−13.87719828237322816234367908278, −12.52266066275554408374720342237, −11.89546949185866168717389197394, −11.08370732871731976402166823692, −8.979653175477284962532040886195, −8.121981335637259796114686103481, −6.81681128465316985589740164654, −6.05276335612152903536262310375, −3.79525083884723156320009140956, −1.38451886126928944133462712852,
3.60377068539049339525285314324, 4.44164479019975164136856998693, 5.99805830909670243724244795208, 7.67645826150369563852884299287, 9.250410626570516648167872405354, 9.859221823955482145131577321417, 11.23282088130062380188988318623, 11.70306156160910528494019548333, 13.37641206156314875400024776204, 14.93845207445335064089232193024
