| L(s) = 1 | + (−0.617 + 1.27i)2-s + (−0.948 − 1.44i)3-s + (−1.23 − 1.57i)4-s + (0.857 − 1.60i)5-s + (2.42 − 0.311i)6-s + (−1.64 − 0.327i)7-s + (2.76 − 0.604i)8-s + (−1.20 + 2.74i)9-s + (1.51 + 2.08i)10-s + (2.14 + 2.61i)11-s + (−1.10 + 3.28i)12-s + (−2.50 − 4.68i)13-s + (1.43 − 1.89i)14-s + (−3.13 + 0.278i)15-s + (−0.937 + 3.88i)16-s + (−1.09 + 0.454i)17-s + ⋯ |
| L(s) = 1 | + (−0.436 + 0.899i)2-s + (−0.547 − 0.836i)3-s + (−0.618 − 0.785i)4-s + (0.383 − 0.717i)5-s + (0.991 − 0.127i)6-s + (−0.621 − 0.123i)7-s + (0.976 − 0.213i)8-s + (−0.400 + 0.916i)9-s + (0.478 + 0.658i)10-s + (0.645 + 0.786i)11-s + (−0.318 + 0.947i)12-s + (−0.694 − 1.29i)13-s + (0.382 − 0.505i)14-s + (−0.810 + 0.0718i)15-s + (−0.234 + 0.972i)16-s + (−0.266 + 0.110i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.694 + 0.719i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.694 + 0.719i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.143720 - 0.338404i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.143720 - 0.338404i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.617 - 1.27i)T \) |
| 3 | \( 1 + (0.948 + 1.44i)T \) |
| good | 5 | \( 1 + (-0.857 + 1.60i)T + (-2.77 - 4.15i)T^{2} \) |
| 7 | \( 1 + (1.64 + 0.327i)T + (6.46 + 2.67i)T^{2} \) |
| 11 | \( 1 + (-2.14 - 2.61i)T + (-2.14 + 10.7i)T^{2} \) |
| 13 | \( 1 + (2.50 + 4.68i)T + (-7.22 + 10.8i)T^{2} \) |
| 17 | \( 1 + (1.09 - 0.454i)T + (12.0 - 12.0i)T^{2} \) |
| 19 | \( 1 + (8.04 + 2.44i)T + (15.7 + 10.5i)T^{2} \) |
| 23 | \( 1 + (6.42 + 4.29i)T + (8.80 + 21.2i)T^{2} \) |
| 29 | \( 1 + (-1.18 + 1.44i)T + (-5.65 - 28.4i)T^{2} \) |
| 31 | \( 1 + (5.85 - 5.85i)T - 31iT^{2} \) |
| 37 | \( 1 + (-6.63 + 2.01i)T + (30.7 - 20.5i)T^{2} \) |
| 41 | \( 1 + (3.28 + 2.19i)T + (15.6 + 37.8i)T^{2} \) |
| 43 | \( 1 + (8.90 + 0.877i)T + (42.1 + 8.38i)T^{2} \) |
| 47 | \( 1 + (-7.69 + 3.18i)T + (33.2 - 33.2i)T^{2} \) |
| 53 | \( 1 + (-2.88 - 3.51i)T + (-10.3 + 51.9i)T^{2} \) |
| 59 | \( 1 + (0.303 - 0.567i)T + (-32.7 - 49.0i)T^{2} \) |
| 61 | \( 1 + (-7.67 + 0.755i)T + (59.8 - 11.9i)T^{2} \) |
| 67 | \( 1 + (0.0194 + 0.197i)T + (-65.7 + 13.0i)T^{2} \) |
| 71 | \( 1 + (6.77 + 1.34i)T + (65.5 + 27.1i)T^{2} \) |
| 73 | \( 1 + (0.818 + 4.11i)T + (-67.4 + 27.9i)T^{2} \) |
| 79 | \( 1 + (-2.06 + 4.97i)T + (-55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + (-15.1 - 4.60i)T + (69.0 + 46.1i)T^{2} \) |
| 89 | \( 1 + (0.0260 + 0.0390i)T + (-34.0 + 82.2i)T^{2} \) |
| 97 | \( 1 + (11.2 + 11.2i)T + 97iT^{2} \) |
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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
−10.69737855456097420508608665129, −10.04785375038206726255469308562, −8.931695668389309891151813313788, −8.150896323013957849211035360311, −7.05719941817561575885757201468, −6.38889301638974513384964772241, −5.43254415727562681672248547351, −4.44842614298090773343697785104, −1.99201024221645030284192284828, −0.28769814768252586531238596753,
2.21333165997250472200578346842, 3.60476080762848455192663048180, 4.38774896963087294370163920993, 6.00611398947101018099467337934, 6.76655151551732808885885333499, 8.387863965761934292213847114519, 9.375669455253227076461341289568, 9.877484890126932407163297905669, 10.74710788190099816366358090831, 11.49729253465343537611970148222
