| L(s) = 1 | + (1.40 − 1.30i)3-s + (2.95 + 0.445i)5-s + (0.339 − 0.588i)7-s + (0.0520 − 0.695i)9-s + (1.98 − 0.955i)11-s + (−0.884 − 2.25i)13-s + (4.74 − 3.23i)15-s + (−2.49 + 0.376i)17-s + (0.122 + 1.63i)19-s + (−0.290 − 1.27i)21-s + (−0.0324 − 0.0221i)23-s + (3.75 + 1.15i)25-s + (2.76 + 3.46i)27-s + (5.57 + 5.17i)29-s + (−8.56 + 2.64i)31-s + ⋯ |
| L(s) = 1 | + (0.813 − 0.755i)3-s + (1.32 + 0.199i)5-s + (0.128 − 0.222i)7-s + (0.0173 − 0.231i)9-s + (0.597 − 0.287i)11-s + (−0.245 − 0.624i)13-s + (1.22 − 0.835i)15-s + (−0.606 + 0.0913i)17-s + (0.0281 + 0.375i)19-s + (−0.0634 − 0.278i)21-s + (−0.00677 − 0.00461i)23-s + (0.750 + 0.231i)25-s + (0.531 + 0.666i)27-s + (1.03 + 0.961i)29-s + (−1.53 + 0.474i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 688 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.781 + 0.623i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 688 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.781 + 0.623i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.31742 - 0.811734i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.31742 - 0.811734i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 43 | \( 1 + (-6.51 - 0.781i)T \) |
| good | 3 | \( 1 + (-1.40 + 1.30i)T + (0.224 - 2.99i)T^{2} \) |
| 5 | \( 1 + (-2.95 - 0.445i)T + (4.77 + 1.47i)T^{2} \) |
| 7 | \( 1 + (-0.339 + 0.588i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.98 + 0.955i)T + (6.85 - 8.60i)T^{2} \) |
| 13 | \( 1 + (0.884 + 2.25i)T + (-9.52 + 8.84i)T^{2} \) |
| 17 | \( 1 + (2.49 - 0.376i)T + (16.2 - 5.01i)T^{2} \) |
| 19 | \( 1 + (-0.122 - 1.63i)T + (-18.7 + 2.83i)T^{2} \) |
| 23 | \( 1 + (0.0324 + 0.0221i)T + (8.40 + 21.4i)T^{2} \) |
| 29 | \( 1 + (-5.57 - 5.17i)T + (2.16 + 28.9i)T^{2} \) |
| 31 | \( 1 + (8.56 - 2.64i)T + (25.6 - 17.4i)T^{2} \) |
| 37 | \( 1 + (5.57 + 9.65i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.555 + 2.43i)T + (-36.9 - 17.7i)T^{2} \) |
| 47 | \( 1 + (7.92 + 3.81i)T + (29.3 + 36.7i)T^{2} \) |
| 53 | \( 1 + (2.23 - 5.70i)T + (-38.8 - 36.0i)T^{2} \) |
| 59 | \( 1 + (2.97 + 3.73i)T + (-13.1 + 57.5i)T^{2} \) |
| 61 | \( 1 + (4.87 + 1.50i)T + (50.4 + 34.3i)T^{2} \) |
| 67 | \( 1 + (0.194 + 2.58i)T + (-66.2 + 9.98i)T^{2} \) |
| 71 | \( 1 + (-7.71 + 5.25i)T + (25.9 - 66.0i)T^{2} \) |
| 73 | \( 1 + (1.43 + 3.66i)T + (-53.5 + 49.6i)T^{2} \) |
| 79 | \( 1 + (3.10 - 5.37i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (3.97 - 3.68i)T + (6.20 - 82.7i)T^{2} \) |
| 89 | \( 1 + (-7.30 + 6.77i)T + (6.65 - 88.7i)T^{2} \) |
| 97 | \( 1 + (0.852 - 0.410i)T + (60.4 - 75.8i)T^{2} \) |
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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
−10.44076157120386906369111446062, −9.278547439879891179368539355608, −8.793869763779394621718218802438, −7.71180757887456573249417130176, −6.93019145659009889269073518937, −6.02876313562710058677938545292, −5.06738990272298237724643649609, −3.49518882945153109043362334246, −2.37304592954263420541814958910, −1.49433391674626667599792992813,
1.77398716195502148280088833338, 2.78313440169099714399309552987, 4.10030381129074945153598108808, 4.95829747996184017974024694192, 6.11391041172188668098464312831, 6.91129134695774319374465035626, 8.333782048784844046342758848879, 9.095231625574826989754262914894, 9.581164889885458084394189193919, 10.17251542442546996838725541243
