| L(s) = 1 | + (−0.784 − 0.166i)2-s + (0.527 + 1.64i)3-s + (−1.23 − 0.551i)4-s + (0.600 + 2.82i)5-s + (−0.138 − 1.38i)6-s + (−1.32 + 0.139i)7-s + (2.17 + 1.58i)8-s + (−2.44 + 1.74i)9-s − 2.31i·10-s + (3.24 + 0.696i)11-s + (0.256 − 2.33i)12-s + (−0.435 − 0.392i)13-s + (1.06 + 0.111i)14-s + (−4.34 + 2.48i)15-s + (0.369 + 0.410i)16-s + (0.407 − 1.25i)17-s + ⋯ |
| L(s) = 1 | + (−0.554 − 0.117i)2-s + (0.304 + 0.952i)3-s + (−0.619 − 0.275i)4-s + (0.268 + 1.26i)5-s + (−0.0566 − 0.564i)6-s + (−0.500 + 0.0526i)7-s + (0.770 + 0.559i)8-s + (−0.814 + 0.580i)9-s − 0.732i·10-s + (0.977 + 0.210i)11-s + (0.0740 − 0.674i)12-s + (−0.120 − 0.108i)13-s + (0.284 + 0.0298i)14-s + (−1.12 + 0.640i)15-s + (0.0922 + 0.102i)16-s + (0.0987 − 0.304i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.163 - 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.163 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.563232 + 0.477562i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.563232 + 0.477562i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.527 - 1.64i)T \) |
| 11 | \( 1 + (-3.24 - 0.696i)T \) |
| good | 2 | \( 1 + (0.784 + 0.166i)T + (1.82 + 0.813i)T^{2} \) |
| 5 | \( 1 + (-0.600 - 2.82i)T + (-4.56 + 2.03i)T^{2} \) |
| 7 | \( 1 + (1.32 - 0.139i)T + (6.84 - 1.45i)T^{2} \) |
| 13 | \( 1 + (0.435 + 0.392i)T + (1.35 + 12.9i)T^{2} \) |
| 17 | \( 1 + (-0.407 + 1.25i)T + (-13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (-4.21 + 5.79i)T + (-5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + (2.62 - 1.51i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (0.136 + 1.29i)T + (-28.3 + 6.02i)T^{2} \) |
| 31 | \( 1 + (-5.05 + 5.61i)T + (-3.24 - 30.8i)T^{2} \) |
| 37 | \( 1 + (3.23 - 2.35i)T + (11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (0.235 - 2.23i)T + (-40.1 - 8.52i)T^{2} \) |
| 43 | \( 1 + (-11.2 - 6.48i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-4.70 - 10.5i)T + (-31.4 + 34.9i)T^{2} \) |
| 53 | \( 1 + (-3.40 + 1.10i)T + (42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (-2.43 + 5.47i)T + (-39.4 - 43.8i)T^{2} \) |
| 61 | \( 1 + (5.87 - 5.29i)T + (6.37 - 60.6i)T^{2} \) |
| 67 | \( 1 + (4.35 + 7.53i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (4.54 + 1.47i)T + (57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (5.16 + 7.10i)T + (-22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (-2.86 + 13.4i)T + (-72.1 - 32.1i)T^{2} \) |
| 83 | \( 1 + (7.20 + 7.99i)T + (-8.67 + 82.5i)T^{2} \) |
| 89 | \( 1 - 8.87iT - 89T^{2} \) |
| 97 | \( 1 + (-0.802 - 0.170i)T + (88.6 + 39.4i)T^{2} \) |
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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
−14.21421384946247784675449605002, −13.54420879091763924471996807249, −11.57718347526433941447560741107, −10.64071473089832126746582153053, −9.663019287917019550716511619397, −9.223407074670858272436459848517, −7.62363594110591885752904353978, −6.09176380514522596340813284151, −4.47736913401930934456788656978, −2.92435818336258629402036041105,
1.19856139223978138488966522121, 3.85093046027946211463582101032, 5.64381243511643221211418789560, 7.14239664019546831890209604221, 8.404573678660574332460021264252, 8.973390976180935054157178728303, 10.00150203488539053235974143737, 12.13069547601931509694501529617, 12.51092692001956868412596248431, 13.65348701924114697391064529845
