| L(s) = 1 | + (−0.841 − 0.540i)2-s + (0.142 + 0.989i)3-s + (0.415 + 0.909i)4-s + (−4.09 − 1.20i)5-s + (0.415 − 0.909i)6-s + (0.654 − 0.755i)7-s + (0.142 − 0.989i)8-s + (−0.959 + 0.281i)9-s + (2.79 + 3.22i)10-s + (−5.36 + 3.44i)11-s + (−0.841 + 0.540i)12-s + (2.29 + 2.64i)13-s + (−0.959 + 0.281i)14-s + (0.607 − 4.22i)15-s + (−0.654 + 0.755i)16-s + (1.08 − 2.37i)17-s + ⋯ |
| L(s) = 1 | + (−0.594 − 0.382i)2-s + (0.0821 + 0.571i)3-s + (0.207 + 0.454i)4-s + (−1.83 − 0.538i)5-s + (0.169 − 0.371i)6-s + (0.247 − 0.285i)7-s + (0.0503 − 0.349i)8-s + (−0.319 + 0.0939i)9-s + (0.884 + 1.02i)10-s + (−1.61 + 1.03i)11-s + (−0.242 + 0.156i)12-s + (0.635 + 0.733i)13-s + (−0.256 + 0.0752i)14-s + (0.156 − 1.09i)15-s + (−0.163 + 0.188i)16-s + (0.262 − 0.575i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.814 + 0.580i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.814 + 0.580i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.612588 - 0.196126i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.612588 - 0.196126i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.841 + 0.540i)T \) |
| 3 | \( 1 + (-0.142 - 0.989i)T \) |
| 7 | \( 1 + (-0.654 + 0.755i)T \) |
| 23 | \( 1 + (-4.77 - 0.426i)T \) |
| good | 5 | \( 1 + (4.09 + 1.20i)T + (4.20 + 2.70i)T^{2} \) |
| 11 | \( 1 + (5.36 - 3.44i)T + (4.56 - 10.0i)T^{2} \) |
| 13 | \( 1 + (-2.29 - 2.64i)T + (-1.85 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-1.08 + 2.37i)T + (-11.1 - 12.8i)T^{2} \) |
| 19 | \( 1 + (1.22 + 2.68i)T + (-12.4 + 14.3i)T^{2} \) |
| 29 | \( 1 + (-1.35 + 2.95i)T + (-18.9 - 21.9i)T^{2} \) |
| 31 | \( 1 + (-0.664 + 4.62i)T + (-29.7 - 8.73i)T^{2} \) |
| 37 | \( 1 + (1.52 - 0.449i)T + (31.1 - 20.0i)T^{2} \) |
| 41 | \( 1 + (1.74 + 0.513i)T + (34.4 + 22.1i)T^{2} \) |
| 43 | \( 1 + (1.23 + 8.62i)T + (-41.2 + 12.1i)T^{2} \) |
| 47 | \( 1 - 11.5T + 47T^{2} \) |
| 53 | \( 1 + (-1.65 + 1.90i)T + (-7.54 - 52.4i)T^{2} \) |
| 59 | \( 1 + (-6.67 - 7.70i)T + (-8.39 + 58.3i)T^{2} \) |
| 61 | \( 1 + (-0.892 + 6.20i)T + (-58.5 - 17.1i)T^{2} \) |
| 67 | \( 1 + (-5.71 - 3.67i)T + (27.8 + 60.9i)T^{2} \) |
| 71 | \( 1 + (0.445 + 0.286i)T + (29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (-0.143 - 0.313i)T + (-47.8 + 55.1i)T^{2} \) |
| 79 | \( 1 + (-2.26 - 2.60i)T + (-11.2 + 78.1i)T^{2} \) |
| 83 | \( 1 + (-2.28 + 0.670i)T + (69.8 - 44.8i)T^{2} \) |
| 89 | \( 1 + (0.0494 + 0.343i)T + (-85.3 + 25.0i)T^{2} \) |
| 97 | \( 1 + (7.10 + 2.08i)T + (81.6 + 52.4i)T^{2} \) |
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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
−10.01259797390564121478520762266, −8.955945375744669043972549079439, −8.400200843237725829540246508987, −7.52814865360408326277714651434, −7.10229209181888151925043261432, −5.17020219299761279600066507544, −4.46927840771074854921800689573, −3.69800375755925357010694116270, −2.49076655301274477715468996070, −0.55695911031021390358492773843,
0.803601984883168645052979324646, 2.79417849941559768435561838335, 3.53188845820445543371317669971, 5.02489795306820312460546145048, 5.98210531105984049683869609083, 7.01647649751256563875186323418, 7.80655410221252412733530634748, 8.256526551702694891696831841569, 8.697204914480224222858078589490, 10.49947838985655004473071708078
