| L(s) = 1 | + (−1.37 − 0.313i)2-s + (−1.73 − 0.0251i)3-s + (1.80 + 0.863i)4-s + (−0.847 + 2.79i)5-s + (2.38 + 0.577i)6-s + (0.874 + 4.39i)7-s + (−2.21 − 1.75i)8-s + (2.99 + 0.0872i)9-s + (2.04 − 3.58i)10-s + (0.263 − 0.0259i)11-s + (−3.10 − 1.54i)12-s + (−1.43 − 4.72i)13-s + (0.170 − 6.33i)14-s + (1.53 − 4.81i)15-s + (2.50 + 3.11i)16-s + (−5.15 − 2.13i)17-s + ⋯ |
| L(s) = 1 | + (−0.975 − 0.221i)2-s + (−0.999 − 0.0145i)3-s + (0.901 + 0.431i)4-s + (−0.378 + 1.24i)5-s + (0.971 + 0.235i)6-s + (0.330 + 1.66i)7-s + (−0.783 − 0.620i)8-s + (0.999 + 0.0290i)9-s + (0.646 − 1.13i)10-s + (0.0795 − 0.00783i)11-s + (−0.895 − 0.444i)12-s + (−0.397 − 1.31i)13-s + (0.0456 − 1.69i)14-s + (0.396 − 1.24i)15-s + (0.626 + 0.779i)16-s + (−1.24 − 0.517i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.968 - 0.249i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.968 - 0.249i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0398849 + 0.314389i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0398849 + 0.314389i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.37 + 0.313i)T \) |
| 3 | \( 1 + (1.73 + 0.0251i)T \) |
| good | 5 | \( 1 + (0.847 - 2.79i)T + (-4.15 - 2.77i)T^{2} \) |
| 7 | \( 1 + (-0.874 - 4.39i)T + (-6.46 + 2.67i)T^{2} \) |
| 11 | \( 1 + (-0.263 + 0.0259i)T + (10.7 - 2.14i)T^{2} \) |
| 13 | \( 1 + (1.43 + 4.72i)T + (-10.8 + 7.22i)T^{2} \) |
| 17 | \( 1 + (5.15 + 2.13i)T + (12.0 + 12.0i)T^{2} \) |
| 19 | \( 1 + (2.20 - 4.11i)T + (-10.5 - 15.7i)T^{2} \) |
| 23 | \( 1 + (-3.95 - 5.91i)T + (-8.80 + 21.2i)T^{2} \) |
| 29 | \( 1 + (-0.738 - 0.0727i)T + (28.4 + 5.65i)T^{2} \) |
| 31 | \( 1 + (1.24 + 1.24i)T + 31iT^{2} \) |
| 37 | \( 1 + (0.796 + 1.49i)T + (-20.5 + 30.7i)T^{2} \) |
| 41 | \( 1 + (-0.0428 - 0.0641i)T + (-15.6 + 37.8i)T^{2} \) |
| 43 | \( 1 + (8.82 + 7.24i)T + (8.38 + 42.1i)T^{2} \) |
| 47 | \( 1 + (0.670 + 0.277i)T + (33.2 + 33.2i)T^{2} \) |
| 53 | \( 1 + (8.08 - 0.795i)T + (51.9 - 10.3i)T^{2} \) |
| 59 | \( 1 + (2.18 - 7.18i)T + (-49.0 - 32.7i)T^{2} \) |
| 61 | \( 1 + (-7.08 + 5.81i)T + (11.9 - 59.8i)T^{2} \) |
| 67 | \( 1 + (1.68 + 2.05i)T + (-13.0 + 65.7i)T^{2} \) |
| 71 | \( 1 + (0.0455 + 0.228i)T + (-65.5 + 27.1i)T^{2} \) |
| 73 | \( 1 + (-3.61 - 0.719i)T + (67.4 + 27.9i)T^{2} \) |
| 79 | \( 1 + (2.23 + 5.40i)T + (-55.8 + 55.8i)T^{2} \) |
| 83 | \( 1 + (0.888 - 1.66i)T + (-46.1 - 69.0i)T^{2} \) |
| 89 | \( 1 + (-0.124 - 0.0830i)T + (34.0 + 82.2i)T^{2} \) |
| 97 | \( 1 + (7.91 - 7.91i)T - 97iT^{2} \) |
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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
−11.50427701795308391788257588755, −10.93245640463752308365712230032, −10.13020346824053331653143074410, −9.117649602998866429603714965110, −8.012481224608297297062522456838, −7.10244183736722925654449986690, −6.20618147359541108955552119343, −5.27122703211662858636115543978, −3.29621097506733728920407441187, −2.12031371851308017686285477420,
0.32761595195382630347824023000, 1.55220774999193481448693390620, 4.39259704748868295811494184161, 4.78829686229164283072038916725, 6.62826133956539733220382941648, 6.94762967878449101474639007154, 8.204444793083308299114285548178, 9.052378804764551110986487639876, 10.05136018829310725570812111301, 11.02618875964679998119570750472
