| L(s) = 1 | + (−2.43 + 0.255i)3-s + (1.56 + 2.71i)5-s + (−7.68 − 1.63i)7-s + (−2.93 + 0.624i)9-s + (−6.11 − 5.50i)11-s + (−7.02 + 15.7i)13-s + (−4.51 − 6.20i)15-s + (−10.6 + 9.56i)17-s + (−12.3 + 5.51i)19-s + (19.1 + 2.01i)21-s + (−10.3 − 3.37i)23-s + (7.58 − 13.1i)25-s + (27.9 − 9.08i)27-s + (11.2 − 15.4i)29-s + (30.8 + 3.48i)31-s + ⋯ |
| L(s) = 1 | + (−0.811 + 0.0853i)3-s + (0.313 + 0.542i)5-s + (−1.09 − 0.233i)7-s + (−0.326 + 0.0693i)9-s + (−0.555 − 0.500i)11-s + (−0.540 + 1.21i)13-s + (−0.300 − 0.413i)15-s + (−0.624 + 0.562i)17-s + (−0.651 + 0.290i)19-s + (0.910 + 0.0957i)21-s + (−0.450 − 0.146i)23-s + (0.303 − 0.525i)25-s + (1.03 − 0.336i)27-s + (0.388 − 0.534i)29-s + (0.993 + 0.112i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.949 - 0.313i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.949 - 0.313i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0445137 + 0.276635i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0445137 + 0.276635i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 31 | \( 1 + (-30.8 - 3.48i)T \) |
| good | 3 | \( 1 + (2.43 - 0.255i)T + (8.80 - 1.87i)T^{2} \) |
| 5 | \( 1 + (-1.56 - 2.71i)T + (-12.5 + 21.6i)T^{2} \) |
| 7 | \( 1 + (7.68 + 1.63i)T + (44.7 + 19.9i)T^{2} \) |
| 11 | \( 1 + (6.11 + 5.50i)T + (12.6 + 120. i)T^{2} \) |
| 13 | \( 1 + (7.02 - 15.7i)T + (-113. - 125. i)T^{2} \) |
| 17 | \( 1 + (10.6 - 9.56i)T + (30.2 - 287. i)T^{2} \) |
| 19 | \( 1 + (12.3 - 5.51i)T + (241. - 268. i)T^{2} \) |
| 23 | \( 1 + (10.3 + 3.37i)T + (427. + 310. i)T^{2} \) |
| 29 | \( 1 + (-11.2 + 15.4i)T + (-259. - 799. i)T^{2} \) |
| 37 | \( 1 + (-24.4 - 14.1i)T + (684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + (3.13 - 29.8i)T + (-1.64e3 - 349. i)T^{2} \) |
| 43 | \( 1 + (2.51 + 5.65i)T + (-1.23e3 + 1.37e3i)T^{2} \) |
| 47 | \( 1 + (56.4 - 41.0i)T + (682. - 2.10e3i)T^{2} \) |
| 53 | \( 1 + (-11.0 - 51.9i)T + (-2.56e3 + 1.14e3i)T^{2} \) |
| 59 | \( 1 + (-2.52 - 24.0i)T + (-3.40e3 + 723. i)T^{2} \) |
| 61 | \( 1 + 100. iT - 3.72e3T^{2} \) |
| 67 | \( 1 + (18.1 + 31.3i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 + (86.9 - 18.4i)T + (4.60e3 - 2.05e3i)T^{2} \) |
| 73 | \( 1 + (-47.6 - 42.8i)T + (557. + 5.29e3i)T^{2} \) |
| 79 | \( 1 + (30.3 - 27.2i)T + (652. - 6.20e3i)T^{2} \) |
| 83 | \( 1 + (16.3 + 1.71i)T + (6.73e3 + 1.43e3i)T^{2} \) |
| 89 | \( 1 + (124. - 40.3i)T + (6.40e3 - 4.65e3i)T^{2} \) |
| 97 | \( 1 + (0.149 + 0.460i)T + (-7.61e3 + 5.53e3i)T^{2} \) |
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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
−13.60723054468993329181332013106, −12.54080071988222042848741489225, −11.48342385133154606267259063142, −10.55046629746733390384164966984, −9.723990879571333047446599896851, −8.310364101334241436433535594883, −6.57242370764995128373031131631, −6.17606924902301929555738832113, −4.49568120875096978294734955621, −2.71212365541967690884908704041,
0.19841098978347737434013360599, 2.82017606135450411946759034902, 4.91075396213208960378060160669, 5.85016043132981680039204041023, 6.97703611096384918025027066581, 8.511979510687571022615422053706, 9.683715473480275657316118490323, 10.57521002495018757871919372535, 11.84294923285120342479826610281, 12.75075862075936519935763567191
