| L(s) = 1 | + (−0.978 − 0.207i)2-s + (1.41 + 2.44i)3-s + (0.913 + 0.406i)4-s + (−0.347 − 3.30i)5-s + (−0.872 − 2.68i)6-s + (−2.60 − 0.480i)7-s + (−0.809 − 0.587i)8-s + (−2.48 + 4.30i)9-s + (−0.347 + 3.30i)10-s + (0.588 − 5.59i)11-s + (0.295 + 2.80i)12-s + (−0.401 − 1.23i)13-s + (2.44 + 1.01i)14-s + (7.59 − 5.51i)15-s + (0.669 + 0.743i)16-s + (0.274 − 2.61i)17-s + ⋯ |
| L(s) = 1 | + (−0.691 − 0.147i)2-s + (0.815 + 1.41i)3-s + (0.456 + 0.203i)4-s + (−0.155 − 1.47i)5-s + (−0.356 − 1.09i)6-s + (−0.983 − 0.181i)7-s + (−0.286 − 0.207i)8-s + (−0.829 + 1.43i)9-s + (−0.109 + 1.04i)10-s + (0.177 − 1.68i)11-s + (0.0852 + 0.810i)12-s + (−0.111 − 0.342i)13-s + (0.653 + 0.270i)14-s + (1.95 − 1.42i)15-s + (0.167 + 0.185i)16-s + (0.0666 − 0.633i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 574 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.327 + 0.944i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 574 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.327 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.767090 - 0.546197i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.767090 - 0.546197i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.978 + 0.207i)T \) |
| 7 | \( 1 + (2.60 + 0.480i)T \) |
| 41 | \( 1 + (0.254 + 6.39i)T \) |
| good | 3 | \( 1 + (-1.41 - 2.44i)T + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (0.347 + 3.30i)T + (-4.89 + 1.03i)T^{2} \) |
| 11 | \( 1 + (-0.588 + 5.59i)T + (-10.7 - 2.28i)T^{2} \) |
| 13 | \( 1 + (0.401 + 1.23i)T + (-10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (-0.274 + 2.61i)T + (-16.6 - 3.53i)T^{2} \) |
| 19 | \( 1 + (4.84 + 5.38i)T + (-1.98 + 18.8i)T^{2} \) |
| 23 | \( 1 + (-5.69 - 1.21i)T + (21.0 + 9.35i)T^{2} \) |
| 29 | \( 1 + (5.64 - 4.10i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (-0.00249 + 0.0237i)T + (-30.3 - 6.44i)T^{2} \) |
| 37 | \( 1 + (-0.592 - 5.63i)T + (-36.1 + 7.69i)T^{2} \) |
| 43 | \( 1 + (2.13 + 6.58i)T + (-34.7 + 25.2i)T^{2} \) |
| 47 | \( 1 + (-7.28 - 1.54i)T + (42.9 + 19.1i)T^{2} \) |
| 53 | \( 1 + (-5.58 - 2.48i)T + (35.4 + 39.3i)T^{2} \) |
| 59 | \( 1 + (5.28 - 5.87i)T + (-6.16 - 58.6i)T^{2} \) |
| 61 | \( 1 + (-6.21 - 6.90i)T + (-6.37 + 60.6i)T^{2} \) |
| 67 | \( 1 + (9.69 + 4.31i)T + (44.8 + 49.7i)T^{2} \) |
| 71 | \( 1 + (3.79 + 2.75i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (1.02 + 1.76i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-4.74 + 8.21i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 6.19T + 83T^{2} \) |
| 89 | \( 1 + (-0.776 - 0.862i)T + (-9.30 + 88.5i)T^{2} \) |
| 97 | \( 1 + (-10.9 + 7.92i)T + (29.9 - 92.2i)T^{2} \) |
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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
−10.42904522428703011966576091738, −9.309820077665474926066190391071, −8.862228627844173830831344487174, −8.648459050481708299558792059022, −7.26015374312364115780421763083, −5.76310445858452992436180667992, −4.76376864300769561711222135889, −3.67141790074604212596799641895, −2.86157421627265300752734634163, −0.58410356107042484697653740415,
1.89400391583954595126136593530, 2.60399709272050565003061538506, 3.79815312978680014212823102879, 6.16281398752177781268661304418, 6.71221460428188909867822436509, 7.29737550607877061377524458581, 7.973628554993636518118036409288, 9.114457882266876917080106327541, 9.905223493417880998014482412213, 10.70755445212025481951478077462
