| L(s) = 1 | + (1.20 + 1.59i)2-s + (1.09 − 1.33i)3-s + (−1.11 + 3.84i)4-s + (−0.743 − 2.45i)5-s + (3.46 + 0.147i)6-s + (2.32 − 11.6i)7-s + (−7.47 + 2.83i)8-s + (−0.585 − 2.94i)9-s + (3.02 − 4.13i)10-s + (13.4 + 1.32i)11-s + (3.92 + 5.71i)12-s + (−22.4 − 6.81i)13-s + (21.4 − 10.3i)14-s + (−4.09 − 1.69i)15-s + (−13.5 − 8.54i)16-s + (−3.07 − 7.42i)17-s + ⋯ |
| L(s) = 1 | + (0.600 + 0.799i)2-s + (0.366 − 0.446i)3-s + (−0.278 + 0.960i)4-s + (−0.148 − 0.490i)5-s + (0.576 + 0.0246i)6-s + (0.331 − 1.66i)7-s + (−0.934 + 0.354i)8-s + (−0.0650 − 0.326i)9-s + (0.302 − 0.413i)10-s + (1.22 + 0.120i)11-s + (0.326 + 0.475i)12-s + (−1.72 − 0.524i)13-s + (1.53 − 0.736i)14-s + (−0.273 − 0.113i)15-s + (−0.845 − 0.534i)16-s + (−0.180 − 0.436i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.800 + 0.599i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.800 + 0.599i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.18423 - 0.727163i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.18423 - 0.727163i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.20 - 1.59i)T \) |
| 3 | \( 1 + (-1.09 + 1.33i)T \) |
| good | 5 | \( 1 + (0.743 + 2.45i)T + (-20.7 + 13.8i)T^{2} \) |
| 7 | \( 1 + (-2.32 + 11.6i)T + (-45.2 - 18.7i)T^{2} \) |
| 11 | \( 1 + (-13.4 - 1.32i)T + (118. + 23.6i)T^{2} \) |
| 13 | \( 1 + (22.4 + 6.81i)T + (140. + 93.8i)T^{2} \) |
| 17 | \( 1 + (3.07 + 7.42i)T + (-204. + 204. i)T^{2} \) |
| 19 | \( 1 + (-22.7 + 12.1i)T + (200. - 300. i)T^{2} \) |
| 23 | \( 1 + (9.90 + 6.62i)T + (202. + 488. i)T^{2} \) |
| 29 | \( 1 + (-23.7 + 2.33i)T + (824. - 164. i)T^{2} \) |
| 31 | \( 1 + (25.1 - 25.1i)T - 961iT^{2} \) |
| 37 | \( 1 + (-21.5 - 11.4i)T + (760. + 1.13e3i)T^{2} \) |
| 41 | \( 1 + (-31.1 - 20.8i)T + (643. + 1.55e3i)T^{2} \) |
| 43 | \( 1 + (-39.2 - 47.8i)T + (-360. + 1.81e3i)T^{2} \) |
| 47 | \( 1 + (-2.72 - 6.58i)T + (-1.56e3 + 1.56e3i)T^{2} \) |
| 53 | \( 1 + (-6.38 - 0.629i)T + (2.75e3 + 548. i)T^{2} \) |
| 59 | \( 1 + (16.1 + 53.3i)T + (-2.89e3 + 1.93e3i)T^{2} \) |
| 61 | \( 1 + (40.6 - 49.5i)T + (-725. - 3.64e3i)T^{2} \) |
| 67 | \( 1 + (60.0 + 49.2i)T + (875. + 4.40e3i)T^{2} \) |
| 71 | \( 1 + (-22.0 - 4.37i)T + (4.65e3 + 1.92e3i)T^{2} \) |
| 73 | \( 1 + (-1.72 + 0.343i)T + (4.92e3 - 2.03e3i)T^{2} \) |
| 79 | \( 1 + (32.0 - 77.4i)T + (-4.41e3 - 4.41e3i)T^{2} \) |
| 83 | \( 1 + (22.5 + 42.2i)T + (-3.82e3 + 5.72e3i)T^{2} \) |
| 89 | \( 1 + (-65.3 - 97.8i)T + (-3.03e3 + 7.31e3i)T^{2} \) |
| 97 | \( 1 + (-30.1 - 30.1i)T + 9.40e3iT^{2} \) |
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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
−11.31157692530494548176757526430, −9.886152308496039719683208076641, −9.021908670361817043134228298868, −7.76927649997641318413867875122, −7.34601518092476723859943033240, −6.52442675306476777657820822892, −4.90539084595348672066246881100, −4.30925804960067419382416102259, −2.97570744275206395516850175844, −0.834085994038602198167952951393,
1.95212133356580428892479615657, 2.88856392151460851159689737426, 4.08866827603262718488649405739, 5.20809724546232929593804472804, 6.07954597440378907641655893015, 7.45909725306442126202008642677, 9.025790881923826684046261779147, 9.313209949898174511038442706168, 10.34379877198199671571261960284, 11.55756839738853075713425182890
