| L(s) = 1 | + (0.173 − 0.984i)2-s + (−0.111 − 1.72i)3-s + (−0.939 − 0.342i)4-s + (−0.359 − 0.989i)5-s + (−1.72 − 0.190i)6-s + (−2.21 + 1.44i)7-s + (−0.5 + 0.866i)8-s + (−2.97 + 0.384i)9-s + (−1.03 + 0.182i)10-s + 1.87i·11-s + (−0.486 + 1.66i)12-s + (0.976 − 0.172i)13-s + (1.04 + 2.43i)14-s + (−1.66 + 0.732i)15-s + (0.766 + 0.642i)16-s + (−3.66 + 4.36i)17-s + ⋯ |
| L(s) = 1 | + (0.122 − 0.696i)2-s + (−0.0642 − 0.997i)3-s + (−0.469 − 0.171i)4-s + (−0.160 − 0.442i)5-s + (−0.702 − 0.0778i)6-s + (−0.836 + 0.547i)7-s + (−0.176 + 0.306i)8-s + (−0.991 + 0.128i)9-s + (−0.327 + 0.0577i)10-s + 0.564i·11-s + (−0.140 + 0.479i)12-s + (0.270 − 0.0477i)13-s + (0.278 + 0.649i)14-s + (−0.431 + 0.189i)15-s + (0.191 + 0.160i)16-s + (−0.889 + 1.05i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.313 - 0.949i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.313 - 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0900133 + 0.0650643i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0900133 + 0.0650643i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.173 + 0.984i)T \) |
| 3 | \( 1 + (0.111 + 1.72i)T \) |
| 7 | \( 1 + (2.21 - 1.44i)T \) |
| 19 | \( 1 + (4.34 - 0.326i)T \) |
| good | 5 | \( 1 + (0.359 + 0.989i)T + (-3.83 + 3.21i)T^{2} \) |
| 11 | \( 1 - 1.87iT - 11T^{2} \) |
| 13 | \( 1 + (-0.976 + 0.172i)T + (12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (3.66 - 4.36i)T + (-2.95 - 16.7i)T^{2} \) |
| 23 | \( 1 + (-0.876 + 0.154i)T + (21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (-0.502 - 0.182i)T + (22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (3.84 + 2.21i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (3.78 + 2.18i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (2.05 - 11.6i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (5.48 + 4.60i)T + (7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (-4.05 - 4.83i)T + (-8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + (2.38 + 0.867i)T + (40.6 + 34.0i)T^{2} \) |
| 59 | \( 1 + (6.88 + 5.77i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (1.92 + 10.9i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-5.33 + 0.940i)T + (62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (-5.23 - 4.39i)T + (12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 + (-0.322 + 0.117i)T + (55.9 - 46.9i)T^{2} \) |
| 79 | \( 1 + (7.89 - 9.40i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (11.1 - 6.42i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (6.57 + 2.39i)T + (68.1 + 57.2i)T^{2} \) |
| 97 | \( 1 + (-1.99 - 5.49i)T + (-74.3 + 62.3i)T^{2} \) |
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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
−10.62429175215909714022250096988, −9.541483260029771953659891592805, −8.692701213572792826952784957763, −8.144955214568003044627945391360, −6.75820795302020922564874875469, −6.20083742028032668530386260904, −5.06523359942719690289117906591, −3.88663269258896918814742857920, −2.64403402307349320558758874589, −1.66337663459224559019402816213,
0.05153899246614330843250115996, 2.91839455442170584834671807590, 3.74559176044264806373432173107, 4.66925358106362805380615204919, 5.70164382354802751489224277297, 6.63791566253558885461223168858, 7.27724218600519821493233184082, 8.698031567895106888833892641601, 9.035461906170372590302029563504, 10.14675938718666125952136205823
