| 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
−11.19634896268834436145685250051, −10.33871717731306619996377728107, −9.606961195716689179918915065625, −8.338222985250277462501086070277, −7.74428115719068854540744649858, −6.47196259225973339750130289176, −5.81251422187098768136605142057, −4.93294605478421144541209658154, −3.10473559203660717679556511851, −2.31967494369212365888425815523,
0.05183814737396224136893635827, 2.11410328393162435988285725396, 3.93827702015280505978910726296, 4.47197608162756555381618463411, 5.65476659700531270291047528613, 6.87850587906822563016867886545, 7.40141367457838853379493025992, 8.919641605952537061984977327782, 9.439651543096616540894087937006, 10.57956359843644923132501168616
