| L(s) = 1 | + (5.05 + 1.22i)3-s + (11.1 − 4.06i)5-s + (−4.28 − 24.2i)7-s + (24.0 + 12.3i)9-s + (−9.00 − 3.27i)11-s + (6.18 + 5.18i)13-s + (61.3 − 6.89i)15-s + (−16.8 + 29.1i)17-s + (57.3 + 99.3i)19-s + (8.01 − 127. i)21-s + (29.4 − 167. i)23-s + (12.3 − 10.3i)25-s + (106. + 91.6i)27-s + (−1.33 + 1.11i)29-s + (−26.8 + 152. i)31-s + ⋯ |
| L(s) = 1 | + (0.972 + 0.234i)3-s + (0.998 − 0.363i)5-s + (−0.231 − 1.31i)7-s + (0.889 + 0.456i)9-s + (−0.246 − 0.0898i)11-s + (0.131 + 0.110i)13-s + (1.05 − 0.118i)15-s + (−0.239 + 0.415i)17-s + (0.692 + 1.19i)19-s + (0.0833 − 1.32i)21-s + (0.267 − 1.51i)23-s + (0.0984 − 0.0826i)25-s + (0.757 + 0.652i)27-s + (−0.00851 + 0.00714i)29-s + (−0.155 + 0.883i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.954 + 0.299i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.954 + 0.299i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.39029 - 0.366400i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.39029 - 0.366400i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-5.05 - 1.22i)T \) |
| good | 5 | \( 1 + (-11.1 + 4.06i)T + (95.7 - 80.3i)T^{2} \) |
| 7 | \( 1 + (4.28 + 24.2i)T + (-322. + 117. i)T^{2} \) |
| 11 | \( 1 + (9.00 + 3.27i)T + (1.01e3 + 855. i)T^{2} \) |
| 13 | \( 1 + (-6.18 - 5.18i)T + (381. + 2.16e3i)T^{2} \) |
| 17 | \( 1 + (16.8 - 29.1i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-57.3 - 99.3i)T + (-3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-29.4 + 167. i)T + (-1.14e4 - 4.16e3i)T^{2} \) |
| 29 | \( 1 + (1.33 - 1.11i)T + (4.23e3 - 2.40e4i)T^{2} \) |
| 31 | \( 1 + (26.8 - 152. i)T + (-2.79e4 - 1.01e4i)T^{2} \) |
| 37 | \( 1 + (143. - 247. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + (-93.7 - 78.6i)T + (1.19e4 + 6.78e4i)T^{2} \) |
| 43 | \( 1 + (328. + 119. i)T + (6.09e4 + 5.11e4i)T^{2} \) |
| 47 | \( 1 + (40.1 + 227. i)T + (-9.75e4 + 3.55e4i)T^{2} \) |
| 53 | \( 1 + 647.T + 1.48e5T^{2} \) |
| 59 | \( 1 + (824. - 299. i)T + (1.57e5 - 1.32e5i)T^{2} \) |
| 61 | \( 1 + (-126. - 715. i)T + (-2.13e5 + 7.76e4i)T^{2} \) |
| 67 | \( 1 + (210. + 176. i)T + (5.22e4 + 2.96e5i)T^{2} \) |
| 71 | \( 1 + (-392. + 679. i)T + (-1.78e5 - 3.09e5i)T^{2} \) |
| 73 | \( 1 + (-176. - 305. i)T + (-1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-820. + 688. i)T + (8.56e4 - 4.85e5i)T^{2} \) |
| 83 | \( 1 + (-1.01e3 + 849. i)T + (9.92e4 - 5.63e5i)T^{2} \) |
| 89 | \( 1 + (-15.8 - 27.3i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + (-64.2 - 23.3i)T + (6.99e5 + 5.86e5i)T^{2} \) |
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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
−13.52740144115245396715010935622, −12.48862681707896267393391650887, −10.54293650639124825091544927634, −10.04914740887955893861747901360, −8.902494768590644197283420652646, −7.78056373955910901139400126866, −6.48745984819829923389609770011, −4.77583891266784672987996161935, −3.39186735044263624386837568975, −1.56693184698929101996601837684,
2.04538706721881350958063989857, 3.09496014771888484584710602762, 5.28168115777237557255395216737, 6.52508884161247401963523965988, 7.83468183257910522108846941382, 9.329665071387239307467256061644, 9.485473112497362107825665451033, 11.18107882141177513401861602281, 12.49837754659838228653534802293, 13.42540747573259774305736736792
