| L(s) = 1 | + (−0.654 + 0.755i)3-s + (0.140 − 0.978i)5-s + (−1.46 − 3.19i)7-s + (−0.142 − 0.989i)9-s + (−2.56 + 0.754i)11-s + (−2.78 + 6.08i)13-s + (0.647 + 0.746i)15-s + (−2.27 + 1.46i)17-s + (−3.51 − 2.25i)19-s + (3.37 + 0.991i)21-s + (−4.79 − 0.0536i)23-s + (3.86 + 1.13i)25-s + (0.841 + 0.540i)27-s + (−6.35 + 4.08i)29-s + (−1.84 − 2.12i)31-s + ⋯ |
| L(s) = 1 | + (−0.378 + 0.436i)3-s + (0.0628 − 0.437i)5-s + (−0.552 − 1.20i)7-s + (−0.0474 − 0.329i)9-s + (−0.774 + 0.227i)11-s + (−0.771 + 1.68i)13-s + (0.167 + 0.192i)15-s + (−0.551 + 0.354i)17-s + (−0.806 − 0.518i)19-s + (0.736 + 0.216i)21-s + (−0.999 − 0.0111i)23-s + (0.772 + 0.226i)25-s + (0.161 + 0.104i)27-s + (−1.18 + 0.758i)29-s + (−0.331 − 0.382i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.988 + 0.154i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.988 + 0.154i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.00729544 - 0.0941345i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00729544 - 0.0941345i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.654 - 0.755i)T \) |
| 23 | \( 1 + (4.79 + 0.0536i)T \) |
| good | 5 | \( 1 + (-0.140 + 0.978i)T + (-4.79 - 1.40i)T^{2} \) |
| 7 | \( 1 + (1.46 + 3.19i)T + (-4.58 + 5.29i)T^{2} \) |
| 11 | \( 1 + (2.56 - 0.754i)T + (9.25 - 5.94i)T^{2} \) |
| 13 | \( 1 + (2.78 - 6.08i)T + (-8.51 - 9.82i)T^{2} \) |
| 17 | \( 1 + (2.27 - 1.46i)T + (7.06 - 15.4i)T^{2} \) |
| 19 | \( 1 + (3.51 + 2.25i)T + (7.89 + 17.2i)T^{2} \) |
| 29 | \( 1 + (6.35 - 4.08i)T + (12.0 - 26.3i)T^{2} \) |
| 31 | \( 1 + (1.84 + 2.12i)T + (-4.41 + 30.6i)T^{2} \) |
| 37 | \( 1 + (1.16 + 8.11i)T + (-35.5 + 10.4i)T^{2} \) |
| 41 | \( 1 + (-1.44 + 10.0i)T + (-39.3 - 11.5i)T^{2} \) |
| 43 | \( 1 + (-5.00 + 5.77i)T + (-6.11 - 42.5i)T^{2} \) |
| 47 | \( 1 + 3.24T + 47T^{2} \) |
| 53 | \( 1 + (-5.25 - 11.5i)T + (-34.7 + 40.0i)T^{2} \) |
| 59 | \( 1 + (4.84 - 10.6i)T + (-38.6 - 44.5i)T^{2} \) |
| 61 | \( 1 + (7.57 + 8.73i)T + (-8.68 + 60.3i)T^{2} \) |
| 67 | \( 1 + (-4.87 - 1.43i)T + (56.3 + 36.2i)T^{2} \) |
| 71 | \( 1 + (9.33 + 2.74i)T + (59.7 + 38.3i)T^{2} \) |
| 73 | \( 1 + (-11.2 - 7.21i)T + (30.3 + 66.4i)T^{2} \) |
| 79 | \( 1 + (0.111 - 0.243i)T + (-51.7 - 59.7i)T^{2} \) |
| 83 | \( 1 + (-0.0290 - 0.202i)T + (-79.6 + 23.3i)T^{2} \) |
| 89 | \( 1 + (0.0502 - 0.0580i)T + (-12.6 - 88.0i)T^{2} \) |
| 97 | \( 1 + (-0.758 + 5.27i)T + (-93.0 - 27.3i)T^{2} \) |
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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
−10.57956359843644923132501168616, −9.439651543096616540894087937006, −8.919641605952537061984977327782, −7.40141367457838853379493025992, −6.87850587906822563016867886545, −5.65476659700531270291047528613, −4.47197608162756555381618463411, −3.93827702015280505978910726296, −2.11410328393162435988285725396, −0.05183814737396224136893635827,
2.31967494369212365888425815523, 3.10473559203660717679556511851, 4.93294605478421144541209658154, 5.81251422187098768136605142057, 6.47196259225973339750130289176, 7.74428115719068854540744649858, 8.338222985250277462501086070277, 9.606961195716689179918915065625, 10.33871717731306619996377728107, 11.19634896268834436145685250051
