| L(s) = 1 | + (−2.72 + 1.25i)3-s + (0.340 + 0.590i)5-s + (−2.79 − 4.84i)7-s + (5.83 − 6.85i)9-s + (7.44 + 12.9i)11-s − 2.36i·13-s + (−1.67 − 1.17i)15-s + (−8.72 + 15.1i)17-s + (−17.2 − 8.02i)19-s + (13.7 + 9.67i)21-s − 10.4·23-s + (12.2 − 21.2i)25-s + (−7.27 + 26.0i)27-s + (−37.6 − 21.7i)29-s + (28.9 + 16.7i)31-s + ⋯ |
| L(s) = 1 | + (−0.907 + 0.419i)3-s + (0.0681 + 0.118i)5-s + (−0.399 − 0.691i)7-s + (0.648 − 0.761i)9-s + (0.677 + 1.17i)11-s − 0.182i·13-s + (−0.111 − 0.0785i)15-s + (−0.513 + 0.889i)17-s + (−0.906 − 0.422i)19-s + (0.652 + 0.460i)21-s − 0.454·23-s + (0.490 − 0.849i)25-s + (−0.269 + 0.962i)27-s + (−1.29 − 0.749i)29-s + (0.934 + 0.539i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.176 + 0.984i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.176 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.7751041307\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7751041307\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (2.72 - 1.25i)T \) |
| 19 | \( 1 + (17.2 + 8.02i)T \) |
| good | 5 | \( 1 + (-0.340 - 0.590i)T + (-12.5 + 21.6i)T^{2} \) |
| 7 | \( 1 + (2.79 + 4.84i)T + (-24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + (-7.44 - 12.9i)T + (-60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + 2.36iT - 169T^{2} \) |
| 17 | \( 1 + (8.72 - 15.1i)T + (-144.5 - 250. i)T^{2} \) |
| 23 | \( 1 + 10.4T + 529T^{2} \) |
| 29 | \( 1 + (37.6 + 21.7i)T + (420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (-28.9 - 16.7i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + 30.3iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (-48.7 + 28.1i)T + (840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + 62.4T + 1.84e3T^{2} \) |
| 47 | \( 1 + (-43.3 + 75.0i)T + (-1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + (-43.7 + 25.2i)T + (1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + (-13.9 + 8.02i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-0.219 + 0.380i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + 41.0iT - 4.48e3T^{2} \) |
| 71 | \( 1 + (-62.9 - 36.3i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (-38.3 + 66.4i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 - 120. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (53.7 + 93.0i)T + (-3.44e3 + 5.96e3i)T^{2} \) |
| 89 | \( 1 + (122. - 70.7i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + 171. iT - 9.40e3T^{2} \) |
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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
−10.22247838529337843485060963658, −9.508229113786100660938239453190, −8.441059459680844982618973450640, −7.04322543097805952267965754992, −6.64976551535821425072459187678, −5.60956021408281744226891684196, −4.34457992491349043188196293948, −3.90899652566983189274352840851, −2.01583143863459591269473371773, −0.35106265752850875532833241426,
1.16523881106884821307995621871, 2.62643921009084797325670903704, 4.05932922839304810206400790056, 5.24571195657237302518059135178, 6.08447472684586634472907785675, 6.65715643518607290629977982460, 7.79554756613023953727551311722, 8.837858781053868010718754239819, 9.530989981251643719951058440223, 10.70610355691614678027408240534
