| L(s) = 1 | + (−2.56 − 1.56i)3-s + (−1.88 − 3.26i)5-s + (−0.501 − 0.868i)7-s + (4.12 + 7.99i)9-s + (3.99 + 6.92i)11-s − 2.03i·13-s + (−0.268 + 11.3i)15-s + (−9.87 + 17.0i)17-s + (13.5 + 13.3i)19-s + (−0.0715 + 3.00i)21-s + 10.1·23-s + (5.38 − 9.33i)25-s + (1.92 − 26.9i)27-s + (8.88 + 5.12i)29-s + (−34.4 − 19.8i)31-s + ⋯ |
| L(s) = 1 | + (−0.853 − 0.520i)3-s + (−0.377 − 0.653i)5-s + (−0.0716 − 0.124i)7-s + (0.458 + 0.888i)9-s + (0.363 + 0.629i)11-s − 0.156i·13-s + (−0.0179 + 0.754i)15-s + (−0.580 + 1.00i)17-s + (0.713 + 0.700i)19-s + (−0.00340 + 0.143i)21-s + 0.439·23-s + (0.215 − 0.373i)25-s + (0.0712 − 0.997i)27-s + (0.306 + 0.176i)29-s + (−1.11 − 0.640i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.851 + 0.523i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.851 + 0.523i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.173141596\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.173141596\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (2.56 + 1.56i)T \) |
| 19 | \( 1 + (-13.5 - 13.3i)T \) |
| good | 5 | \( 1 + (1.88 + 3.26i)T + (-12.5 + 21.6i)T^{2} \) |
| 7 | \( 1 + (0.501 + 0.868i)T + (-24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + (-3.99 - 6.92i)T + (-60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + 2.03iT - 169T^{2} \) |
| 17 | \( 1 + (9.87 - 17.0i)T + (-144.5 - 250. i)T^{2} \) |
| 23 | \( 1 - 10.1T + 529T^{2} \) |
| 29 | \( 1 + (-8.88 - 5.12i)T + (420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (34.4 + 19.8i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 - 34.3iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (-60.5 + 34.9i)T + (840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 - 12.8T + 1.84e3T^{2} \) |
| 47 | \( 1 + (17.4 - 30.3i)T + (-1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + (-77.7 + 44.9i)T + (1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + (50.2 - 29.0i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-14.6 + 25.3i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + 95.7iT - 4.48e3T^{2} \) |
| 71 | \( 1 + (-89.8 - 51.8i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (-28.1 + 48.7i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 - 35.8iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (-40.7 - 70.6i)T + (-3.44e3 + 5.96e3i)T^{2} \) |
| 89 | \( 1 + (-84.1 + 48.5i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + 170. iT - 9.40e3T^{2} \) |
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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
−10.36259265061534984029940042411, −9.367484986833192800805911729322, −8.325702743001849393423844169478, −7.52689129955648564025279183765, −6.64763923969534000510499346325, −5.71441101019705999685056175080, −4.76645611793560693777773771584, −3.85413987594624592725746863611, −2.00277576228705661151975702735, −0.77934529350125908519550202003,
0.76686816163492154474152178565, 2.82207644132833579387458393884, 3.84267936848013690408338074241, 4.91999851625226862082935093533, 5.82771911160609602256291406195, 6.84118490787851713985123706508, 7.41670176100490384488003831433, 8.982461939195395091826037160575, 9.409166833084175232325132601401, 10.64814972451070814814747455649
