| L(s) = 1 | + (1.27 − 0.610i)2-s + (−0.995 + 0.0980i)3-s + (1.25 − 1.55i)4-s + (−1.28 + 2.39i)5-s + (−1.20 + 0.732i)6-s + (0.997 − 5.01i)7-s + (0.649 − 2.75i)8-s + (0.980 − 0.195i)9-s + (−0.171 + 3.84i)10-s + (0.439 − 0.360i)11-s + (−1.09 + 1.67i)12-s + (3.44 − 1.84i)13-s + (−1.78 − 7.00i)14-s + (1.04 − 2.51i)15-s + (−0.852 − 3.90i)16-s + (−0.200 − 0.484i)17-s + ⋯ |
| L(s) = 1 | + (0.902 − 0.431i)2-s + (−0.574 + 0.0565i)3-s + (0.627 − 0.778i)4-s + (−0.573 + 1.07i)5-s + (−0.493 + 0.299i)6-s + (0.377 − 1.89i)7-s + (0.229 − 0.973i)8-s + (0.326 − 0.0650i)9-s + (−0.0540 + 1.21i)10-s + (0.132 − 0.108i)11-s + (−0.316 + 0.482i)12-s + (0.955 − 0.510i)13-s + (−0.478 − 1.87i)14-s + (0.268 − 0.648i)15-s + (−0.213 − 0.977i)16-s + (−0.0486 − 0.117i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.320 + 0.947i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.320 + 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.50720 - 1.08080i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.50720 - 1.08080i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.27 + 0.610i)T \) |
| 3 | \( 1 + (0.995 - 0.0980i)T \) |
| good | 5 | \( 1 + (1.28 - 2.39i)T + (-2.77 - 4.15i)T^{2} \) |
| 7 | \( 1 + (-0.997 + 5.01i)T + (-6.46 - 2.67i)T^{2} \) |
| 11 | \( 1 + (-0.439 + 0.360i)T + (2.14 - 10.7i)T^{2} \) |
| 13 | \( 1 + (-3.44 + 1.84i)T + (7.22 - 10.8i)T^{2} \) |
| 17 | \( 1 + (0.200 + 0.484i)T + (-12.0 + 12.0i)T^{2} \) |
| 19 | \( 1 + (-4.89 - 1.48i)T + (15.7 + 10.5i)T^{2} \) |
| 23 | \( 1 + (4.43 + 2.96i)T + (8.80 + 21.2i)T^{2} \) |
| 29 | \( 1 + (2.77 - 3.38i)T + (-5.65 - 28.4i)T^{2} \) |
| 31 | \( 1 + (-3.56 - 3.56i)T + 31iT^{2} \) |
| 37 | \( 1 + (-0.893 - 2.94i)T + (-30.7 + 20.5i)T^{2} \) |
| 41 | \( 1 + (5.07 - 7.59i)T + (-15.6 - 37.8i)T^{2} \) |
| 43 | \( 1 + (10.6 + 1.04i)T + (42.1 + 8.38i)T^{2} \) |
| 47 | \( 1 + (-4.81 + 1.99i)T + (33.2 - 33.2i)T^{2} \) |
| 53 | \( 1 + (0.501 + 0.611i)T + (-10.3 + 51.9i)T^{2} \) |
| 59 | \( 1 + (-9.69 - 5.18i)T + (32.7 + 49.0i)T^{2} \) |
| 61 | \( 1 + (0.0767 + 0.779i)T + (-59.8 + 11.9i)T^{2} \) |
| 67 | \( 1 + (-1.28 - 13.0i)T + (-65.7 + 13.0i)T^{2} \) |
| 71 | \( 1 + (-10.4 - 2.07i)T + (65.5 + 27.1i)T^{2} \) |
| 73 | \( 1 + (0.349 + 1.75i)T + (-67.4 + 27.9i)T^{2} \) |
| 79 | \( 1 + (-8.80 - 3.64i)T + (55.8 + 55.8i)T^{2} \) |
| 83 | \( 1 + (-1.74 + 5.75i)T + (-69.0 - 46.1i)T^{2} \) |
| 89 | \( 1 + (1.79 - 1.19i)T + (34.0 - 82.2i)T^{2} \) |
| 97 | \( 1 + (-8.59 - 8.59i)T + 97iT^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.31789836859088798799969169715, −10.34606485291413197298368363123, −10.18442233399103791532519928346, −7.995453389915012997827849901125, −7.05675344918386223220811426980, −6.46694963827727388598605978556, −5.10108574788155620696142257065, −3.93346355475211117598521195197, −3.33383270397556226040640179437, −1.13918643783380467577637526202,
1.98461481746937226945969070783, 3.73421344645439339876065286595, 4.92071855133622355262289970290, 5.55423316848137906959784242143, 6.40089429239863334847404839266, 7.85141598912988524463435969213, 8.541850868228925797890905045919, 9.424332034856795067178303019610, 11.26246798753519741940910846664, 11.85044860391393203942013079958
