L(s) = 1 | + (−1.5 + 2.59i)2-s + (−0.5 − 0.866i)4-s + (2.5 + 4.33i)5-s + (−10 + 17.3i)7-s − 21·8-s − 15.0·10-s + (12 − 20.7i)11-s + (−37 − 64.0i)13-s + (−30.0 − 51.9i)14-s + (35.5 − 61.4i)16-s + 54·17-s − 124·19-s + (2.50 − 4.33i)20-s + (36 + 62.3i)22-s + (60 + 103. i)23-s + ⋯ |
L(s) = 1 | + (−0.530 + 0.918i)2-s + (−0.0625 − 0.108i)4-s + (0.223 + 0.387i)5-s + (−0.539 + 0.935i)7-s − 0.928·8-s − 0.474·10-s + (0.328 − 0.569i)11-s + (−0.789 − 1.36i)13-s + (−0.572 − 0.991i)14-s + (0.554 − 0.960i)16-s + 0.770·17-s − 1.49·19-s + (0.0279 − 0.0484i)20-s + (0.348 + 0.604i)22-s + (0.543 + 0.942i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.4179052145\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4179052145\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (-2.5 - 4.33i)T \) |
good | 2 | \( 1 + (1.5 - 2.59i)T + (-4 - 6.92i)T^{2} \) |
| 7 | \( 1 + (10 - 17.3i)T + (-171.5 - 297. i)T^{2} \) |
| 11 | \( 1 + (-12 + 20.7i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (37 + 64.0i)T + (-1.09e3 + 1.90e3i)T^{2} \) |
| 17 | \( 1 - 54T + 4.91e3T^{2} \) |
| 19 | \( 1 + 124T + 6.85e3T^{2} \) |
| 23 | \( 1 + (-60 - 103. i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-39 + 67.5i)T + (-1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 + (100 + 173. i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + 70T + 5.06e4T^{2} \) |
| 41 | \( 1 + (165 + 285. i)T + (-3.44e4 + 5.96e4i)T^{2} \) |
| 43 | \( 1 + (46 - 79.6i)T + (-3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 + (-12 + 20.7i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 - 450T + 1.48e5T^{2} \) |
| 59 | \( 1 + (12 + 20.7i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-161 + 278. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-98 - 169. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + 288T + 3.57e5T^{2} \) |
| 73 | \( 1 + 430T + 3.89e5T^{2} \) |
| 79 | \( 1 + (-260 + 450. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + (78 - 135. i)T + (-2.85e5 - 4.95e5i)T^{2} \) |
| 89 | \( 1 - 1.02e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + (-143 + 247. i)T + (-4.56e5 - 7.90e5i)T^{2} \) |
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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
−10.50052817743630538045656477602, −9.589098157379380178436480150357, −8.769200870634848716516728820025, −7.939304051508737282863222268446, −7.03042738772054743014911304056, −6.00094508133569028804001667864, −5.48357609648767074543920883998, −3.45900330764592845205483565608, −2.50272903311968799131390611449, −0.17161051601563216432188918097,
1.23865926332527884056524629186, 2.34432299216424244253065808139, 3.78692971906753789824311146710, 4.86925689177323443874517668201, 6.44220325310688351622540790764, 7.05442837896064095777398280237, 8.602992565046768360185899042139, 9.321144161951093357792947208657, 10.17102159868620546083330937479, 10.64242540460174877212074903316