| L(s) = 1 | + (0.263 − 3.01i)2-s + (0.677 + 2.92i)3-s + (−5.09 − 0.898i)4-s + (0.0911 + 4.99i)5-s + (8.99 − 1.27i)6-s + (5.64 + 3.95i)7-s + (−0.921 + 3.43i)8-s + (−8.08 + 3.95i)9-s + (15.1 + 1.04i)10-s + (17.8 + 6.50i)11-s + (−0.825 − 15.5i)12-s + (−1.37 − 15.6i)13-s + (13.4 − 15.9i)14-s + (−14.5 + 3.65i)15-s + (−9.32 − 3.39i)16-s + (5.77 + 21.5i)17-s + ⋯ |
| L(s) = 1 | + (0.131 − 1.50i)2-s + (0.225 + 0.974i)3-s + (−1.27 − 0.224i)4-s + (0.0182 + 0.999i)5-s + (1.49 − 0.212i)6-s + (0.806 + 0.564i)7-s + (−0.115 + 0.429i)8-s + (−0.898 + 0.439i)9-s + (1.51 + 0.104i)10-s + (1.62 + 0.591i)11-s + (−0.0687 − 1.29i)12-s + (−0.105 − 1.20i)13-s + (0.958 − 1.14i)14-s + (−0.969 + 0.243i)15-s + (−0.582 − 0.212i)16-s + (0.339 + 1.26i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.949 + 0.312i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.949 + 0.312i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.67767 - 0.268686i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.67767 - 0.268686i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.677 - 2.92i)T \) |
| 5 | \( 1 + (-0.0911 - 4.99i)T \) |
| good | 2 | \( 1 + (-0.263 + 3.01i)T + (-3.93 - 0.694i)T^{2} \) |
| 7 | \( 1 + (-5.64 - 3.95i)T + (16.7 + 46.0i)T^{2} \) |
| 11 | \( 1 + (-17.8 - 6.50i)T + (92.6 + 77.7i)T^{2} \) |
| 13 | \( 1 + (1.37 + 15.6i)T + (-166. + 29.3i)T^{2} \) |
| 17 | \( 1 + (-5.77 - 21.5i)T + (-250. + 144.5i)T^{2} \) |
| 19 | \( 1 + (2.26 - 1.30i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (-2.27 + 1.59i)T + (180. - 497. i)T^{2} \) |
| 29 | \( 1 + (13.8 + 16.4i)T + (-146. + 828. i)T^{2} \) |
| 31 | \( 1 + (-8.08 + 45.8i)T + (-903. - 328. i)T^{2} \) |
| 37 | \( 1 + (1.91 + 7.14i)T + (-1.18e3 + 684.5i)T^{2} \) |
| 41 | \( 1 + (4.64 + 3.90i)T + (291. + 1.65e3i)T^{2} \) |
| 43 | \( 1 + (-22.7 - 48.6i)T + (-1.18e3 + 1.41e3i)T^{2} \) |
| 47 | \( 1 + (-19.1 - 13.4i)T + (755. + 2.07e3i)T^{2} \) |
| 53 | \( 1 + (-29.7 - 29.7i)T + 2.80e3iT^{2} \) |
| 59 | \( 1 + (37.7 + 103. i)T + (-2.66e3 + 2.23e3i)T^{2} \) |
| 61 | \( 1 + (12.4 + 70.6i)T + (-3.49e3 + 1.27e3i)T^{2} \) |
| 67 | \( 1 + (61.2 - 5.35i)T + (4.42e3 - 779. i)T^{2} \) |
| 71 | \( 1 + (-12.8 + 22.2i)T + (-2.52e3 - 4.36e3i)T^{2} \) |
| 73 | \( 1 + (-18.0 + 67.4i)T + (-4.61e3 - 2.66e3i)T^{2} \) |
| 79 | \( 1 + (-1.02 - 1.22i)T + (-1.08e3 + 6.14e3i)T^{2} \) |
| 83 | \( 1 + (58.6 + 5.12i)T + (6.78e3 + 1.19e3i)T^{2} \) |
| 89 | \( 1 + (45.4 - 26.2i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-50.1 + 23.4i)T + (6.04e3 - 7.20e3i)T^{2} \) |
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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
−12.59662630420856015899374250533, −11.58794194700563075949880883867, −10.97971739800290970965078972058, −10.08205050628452110637932954099, −9.310831611983080185139594206277, −7.958908197383550840313879972023, −6.00561910404088182885085292479, −4.36066507862372293466590522134, −3.40630485404309252178724729782, −2.06168661431968710471289790690,
1.32320952472343254761374607033, 4.26190393888499066079577388046, 5.48931626077571793801911765555, 6.75723139428995854596098740588, 7.41649910147091387752070520682, 8.704642416853275048588382539191, 9.049643411881029165741356130062, 11.45104952700172661600408584177, 12.12326430309251907218934798371, 13.69074725071432380652524763532
