| L(s) = 1 | + (0.0129 − 0.0119i)3-s + (−3.39 − 0.512i)5-s + (0.134 − 0.232i)7-s + (−0.224 + 2.99i)9-s + (2.96 − 1.42i)11-s + (−0.736 − 1.87i)13-s + (−0.0500 + 0.0341i)15-s + (6.37 − 0.960i)17-s + (0.449 + 5.99i)19-s + (−0.00105 − 0.00462i)21-s + (1.83 + 1.25i)23-s + (6.50 + 2.00i)25-s + (0.0659 + 0.0827i)27-s + (1.75 + 1.63i)29-s + (4.93 − 1.52i)31-s + ⋯ |
| L(s) = 1 | + (0.00746 − 0.00692i)3-s + (−1.51 − 0.229i)5-s + (0.0508 − 0.0880i)7-s + (−0.0747 + 0.997i)9-s + (0.894 − 0.430i)11-s + (−0.204 − 0.520i)13-s + (−0.0129 + 0.00881i)15-s + (1.54 − 0.233i)17-s + (0.103 + 1.37i)19-s + (−0.000230 − 0.00100i)21-s + (0.382 + 0.260i)23-s + (1.30 + 0.401i)25-s + (0.0126 + 0.0159i)27-s + (0.326 + 0.303i)29-s + (0.887 − 0.273i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 688 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.936 - 0.349i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 688 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.936 - 0.349i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.17521 + 0.211971i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.17521 + 0.211971i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 43 | \( 1 + (-4.01 + 5.18i)T \) |
| good | 3 | \( 1 + (-0.0129 + 0.0119i)T + (0.224 - 2.99i)T^{2} \) |
| 5 | \( 1 + (3.39 + 0.512i)T + (4.77 + 1.47i)T^{2} \) |
| 7 | \( 1 + (-0.134 + 0.232i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.96 + 1.42i)T + (6.85 - 8.60i)T^{2} \) |
| 13 | \( 1 + (0.736 + 1.87i)T + (-9.52 + 8.84i)T^{2} \) |
| 17 | \( 1 + (-6.37 + 0.960i)T + (16.2 - 5.01i)T^{2} \) |
| 19 | \( 1 + (-0.449 - 5.99i)T + (-18.7 + 2.83i)T^{2} \) |
| 23 | \( 1 + (-1.83 - 1.25i)T + (8.40 + 21.4i)T^{2} \) |
| 29 | \( 1 + (-1.75 - 1.63i)T + (2.16 + 28.9i)T^{2} \) |
| 31 | \( 1 + (-4.93 + 1.52i)T + (25.6 - 17.4i)T^{2} \) |
| 37 | \( 1 + (-2.63 - 4.56i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (0.643 - 2.81i)T + (-36.9 - 17.7i)T^{2} \) |
| 47 | \( 1 + (-5.31 - 2.56i)T + (29.3 + 36.7i)T^{2} \) |
| 53 | \( 1 + (-2.34 + 5.97i)T + (-38.8 - 36.0i)T^{2} \) |
| 59 | \( 1 + (8.36 + 10.4i)T + (-13.1 + 57.5i)T^{2} \) |
| 61 | \( 1 + (9.50 + 2.93i)T + (50.4 + 34.3i)T^{2} \) |
| 67 | \( 1 + (-0.950 - 12.6i)T + (-66.2 + 9.98i)T^{2} \) |
| 71 | \( 1 + (-4.98 + 3.40i)T + (25.9 - 66.0i)T^{2} \) |
| 73 | \( 1 + (-0.609 - 1.55i)T + (-53.5 + 49.6i)T^{2} \) |
| 79 | \( 1 + (6.97 - 12.0i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-5.60 + 5.19i)T + (6.20 - 82.7i)T^{2} \) |
| 89 | \( 1 + (1.32 - 1.23i)T + (6.65 - 88.7i)T^{2} \) |
| 97 | \( 1 + (-2.36 + 1.13i)T + (60.4 - 75.8i)T^{2} \) |
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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
−10.60804483513900989105493006847, −9.741040246987395592716460576164, −8.486093331538066995881969253218, −7.907414376171898763831953079592, −7.36743770716873139056108698334, −5.98201035402268427562234818221, −4.95012411550679268704023927375, −3.94080071408692075601177538509, −3.07993596227733248990667550851, −1.10291388133809582265701403001,
0.856650557411790896211022983787, 2.94274909130922616581020000259, 3.89188839307130657925119963176, 4.64082712176909475444701003506, 6.13184620647507729316752604475, 7.06415653254272998169518039993, 7.65306550981680142731321263549, 8.790425728972801070078417791015, 9.383695605350239255362795994271, 10.52379927203257639779077853045
