| L(s) = 1 | + (−0.888 + 0.458i)2-s + (0.786 − 0.618i)3-s + (0.580 − 0.814i)4-s + (−0.702 + 0.669i)5-s + (−0.415 + 0.909i)6-s + (−1.56 + 2.13i)7-s + (−0.142 + 0.989i)8-s + (0.235 − 0.971i)9-s + (0.317 − 0.916i)10-s + (−1.62 − 0.836i)11-s + (−0.0475 − 0.998i)12-s + (3.29 + 3.80i)13-s + (0.417 − 2.61i)14-s + (−0.138 + 0.960i)15-s + (−0.327 − 0.945i)16-s + (6.52 − 0.622i)17-s + ⋯ |
| L(s) = 1 | + (−0.628 + 0.324i)2-s + (0.453 − 0.356i)3-s + (0.290 − 0.407i)4-s + (−0.314 + 0.299i)5-s + (−0.169 + 0.371i)6-s + (−0.592 + 0.805i)7-s + (−0.0503 + 0.349i)8-s + (0.0785 − 0.323i)9-s + (0.100 − 0.289i)10-s + (−0.489 − 0.252i)11-s + (−0.0137 − 0.288i)12-s + (0.915 + 1.05i)13-s + (0.111 − 0.698i)14-s + (−0.0356 + 0.247i)15-s + (−0.0817 − 0.236i)16-s + (1.58 − 0.151i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.601 - 0.798i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.601 - 0.798i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.331407 + 0.664645i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.331407 + 0.664645i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.888 - 0.458i)T \) |
| 3 | \( 1 + (-0.786 + 0.618i)T \) |
| 7 | \( 1 + (1.56 - 2.13i)T \) |
| 23 | \( 1 + (2.34 + 4.18i)T \) |
| good | 5 | \( 1 + (0.702 - 0.669i)T + (0.237 - 4.99i)T^{2} \) |
| 11 | \( 1 + (1.62 + 0.836i)T + (6.38 + 8.96i)T^{2} \) |
| 13 | \( 1 + (-3.29 - 3.80i)T + (-1.85 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-6.52 + 0.622i)T + (16.6 - 3.21i)T^{2} \) |
| 19 | \( 1 + (7.47 + 0.714i)T + (18.6 + 3.59i)T^{2} \) |
| 29 | \( 1 + (1.17 - 2.57i)T + (-18.9 - 21.9i)T^{2} \) |
| 31 | \( 1 + (9.48 - 3.79i)T + (22.4 - 21.3i)T^{2} \) |
| 37 | \( 1 + (1.62 - 6.71i)T + (-32.8 - 16.9i)T^{2} \) |
| 41 | \( 1 + (2.31 + 0.680i)T + (34.4 + 22.1i)T^{2} \) |
| 43 | \( 1 + (-0.408 - 2.83i)T + (-41.2 + 12.1i)T^{2} \) |
| 47 | \( 1 + (3.63 - 6.29i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-4.13 - 0.796i)T + (49.2 + 19.6i)T^{2} \) |
| 59 | \( 1 + (2.95 - 8.52i)T + (-46.3 - 36.4i)T^{2} \) |
| 61 | \( 1 + (-3.68 - 2.89i)T + (14.3 + 59.2i)T^{2} \) |
| 67 | \( 1 + (0.167 - 3.50i)T + (-66.6 - 6.36i)T^{2} \) |
| 71 | \( 1 + (-8.46 - 5.43i)T + (29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (6.71 - 9.43i)T + (-23.8 - 68.9i)T^{2} \) |
| 79 | \( 1 + (-8.49 + 1.63i)T + (73.3 - 29.3i)T^{2} \) |
| 83 | \( 1 + (-14.0 + 4.13i)T + (69.8 - 44.8i)T^{2} \) |
| 89 | \( 1 + (6.93 + 2.77i)T + (64.4 + 61.4i)T^{2} \) |
| 97 | \( 1 + (-11.1 - 3.27i)T + (81.6 + 52.4i)T^{2} \) |
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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
−10.21357630609948864355445829280, −9.195984642367436053800623667641, −8.673543271079889168561756988543, −7.947637275113981931291033108776, −6.94136457616463208958856066302, −6.30470677536600337505371390239, −5.39170504866723042317353515554, −3.84410559975510718586077050988, −2.84124923216323206582860365549, −1.63084408874689056960527045206,
0.39186116013345232814309198509, 2.02358441861500236832442867089, 3.54068054735236824312728919181, 3.84495615026392171380332130244, 5.36618562617224213223667312586, 6.39612074725854196074019409274, 7.77071263699566686142378163192, 7.898293191112964936039276810167, 8.948770178502135525105783408689, 9.838946230617371245129560257590
