| L(s) = 1 | + (0.886 − 1.10i)2-s + (0.0980 + 0.995i)3-s + (−0.428 − 1.95i)4-s + (−3.03 − 1.62i)5-s + (1.18 + 0.774i)6-s + (−0.0609 + 0.306i)7-s + (−2.53 − 1.25i)8-s + (−0.980 + 0.195i)9-s + (−4.48 + 1.90i)10-s + (−3.52 − 4.30i)11-s + (1.90 − 0.617i)12-s + (0.800 + 1.49i)13-s + (0.283 + 0.338i)14-s + (1.31 − 3.18i)15-s + (−3.63 + 1.67i)16-s + (−0.777 − 1.87i)17-s + ⋯ |
| L(s) = 1 | + (0.626 − 0.779i)2-s + (0.0565 + 0.574i)3-s + (−0.214 − 0.976i)4-s + (−1.35 − 0.726i)5-s + (0.483 + 0.316i)6-s + (−0.0230 + 0.115i)7-s + (−0.895 − 0.445i)8-s + (−0.326 + 0.0650i)9-s + (−1.41 + 0.603i)10-s + (−1.06 − 1.29i)11-s + (0.549 − 0.178i)12-s + (0.222 + 0.415i)13-s + (0.0758 + 0.0905i)14-s + (0.340 − 0.821i)15-s + (−0.908 + 0.418i)16-s + (−0.188 − 0.455i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.951 + 0.308i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.951 + 0.308i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.148390 - 0.938674i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.148390 - 0.938674i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.886 + 1.10i)T \) |
| 3 | \( 1 + (-0.0980 - 0.995i)T \) |
| good | 5 | \( 1 + (3.03 + 1.62i)T + (2.77 + 4.15i)T^{2} \) |
| 7 | \( 1 + (0.0609 - 0.306i)T + (-6.46 - 2.67i)T^{2} \) |
| 11 | \( 1 + (3.52 + 4.30i)T + (-2.14 + 10.7i)T^{2} \) |
| 13 | \( 1 + (-0.800 - 1.49i)T + (-7.22 + 10.8i)T^{2} \) |
| 17 | \( 1 + (0.777 + 1.87i)T + (-12.0 + 12.0i)T^{2} \) |
| 19 | \( 1 + (-1.40 + 4.63i)T + (-15.7 - 10.5i)T^{2} \) |
| 23 | \( 1 + (-2.04 - 1.36i)T + (8.80 + 21.2i)T^{2} \) |
| 29 | \( 1 + (-2.41 - 1.98i)T + (5.65 + 28.4i)T^{2} \) |
| 31 | \( 1 + (-1.08 - 1.08i)T + 31iT^{2} \) |
| 37 | \( 1 + (5.25 - 1.59i)T + (30.7 - 20.5i)T^{2} \) |
| 41 | \( 1 + (-5.31 + 7.95i)T + (-15.6 - 37.8i)T^{2} \) |
| 43 | \( 1 + (-1.10 + 11.2i)T + (-42.1 - 8.38i)T^{2} \) |
| 47 | \( 1 + (7.30 - 3.02i)T + (33.2 - 33.2i)T^{2} \) |
| 53 | \( 1 + (-1.14 + 0.942i)T + (10.3 - 51.9i)T^{2} \) |
| 59 | \( 1 + (0.278 - 0.520i)T + (-32.7 - 49.0i)T^{2} \) |
| 61 | \( 1 + (-14.9 + 1.47i)T + (59.8 - 11.9i)T^{2} \) |
| 67 | \( 1 + (10.7 - 1.05i)T + (65.7 - 13.0i)T^{2} \) |
| 71 | \( 1 + (-2.97 - 0.591i)T + (65.5 + 27.1i)T^{2} \) |
| 73 | \( 1 + (-1.34 - 6.76i)T + (-67.4 + 27.9i)T^{2} \) |
| 79 | \( 1 + (10.0 + 4.16i)T + (55.8 + 55.8i)T^{2} \) |
| 83 | \( 1 + (3.38 + 1.02i)T + (69.0 + 46.1i)T^{2} \) |
| 89 | \( 1 + (-9.10 + 6.08i)T + (34.0 - 82.2i)T^{2} \) |
| 97 | \( 1 + (1.18 + 1.18i)T + 97iT^{2} \) |
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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
−11.18413633629991643263712639206, −10.33946263603674907980398879672, −8.962345850129865447181505619487, −8.568353546593965369911752700266, −7.16316855848453494954572091375, −5.55212458557834042517813221203, −4.82502233640834620832703605610, −3.80519364907262196221506775225, −2.86753813622426657021395304579, −0.50479550414844983764845176249,
2.69915372545726363303873807709, 3.81141038725759640972357291797, 4.88493004456848483689164927178, 6.21753093613182684641587188628, 7.19615674404493240393960175364, 7.77256362882546385773519014195, 8.357545524133297206676779615125, 10.00477570600101691543826977252, 11.09881605019861055993007390842, 12.01030283811962775259304665356
