| L(s) = 1 | + (0.841 + 0.540i)2-s + (0.142 + 0.989i)3-s + (0.415 + 0.909i)4-s + (−0.571 − 0.167i)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 + (−0.390 − 0.450i)10-s + (4.17 − 2.68i)11-s + (−0.841 + 0.540i)12-s + (3.61 + 4.16i)13-s + (0.959 − 0.281i)14-s + (0.0848 − 0.589i)15-s + (−0.654 + 0.755i)16-s + (2.08 − 4.57i)17-s + ⋯ |
| L(s) = 1 | + (0.594 + 0.382i)2-s + (0.0821 + 0.571i)3-s + (0.207 + 0.454i)4-s + (−0.255 − 0.0750i)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 + (−0.123 − 0.142i)10-s + (1.25 − 0.809i)11-s + (−0.242 + 0.156i)12-s + (1.00 + 1.15i)13-s + (0.256 − 0.0752i)14-s + (0.0218 − 0.152i)15-s + (−0.163 + 0.188i)16-s + (0.506 − 1.10i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.258 - 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.258 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.94331 + 1.49125i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.94331 + 1.49125i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.27 - 4.62i)T \) |
| good | 5 | \( 1 + (0.571 + 0.167i)T + (4.20 + 2.70i)T^{2} \) |
| 11 | \( 1 + (-4.17 + 2.68i)T + (4.56 - 10.0i)T^{2} \) |
| 13 | \( 1 + (-3.61 - 4.16i)T + (-1.85 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-2.08 + 4.57i)T + (-11.1 - 12.8i)T^{2} \) |
| 19 | \( 1 + (-1.40 - 3.07i)T + (-12.4 + 14.3i)T^{2} \) |
| 29 | \( 1 + (2.62 - 5.75i)T + (-18.9 - 21.9i)T^{2} \) |
| 31 | \( 1 + (1.02 - 7.15i)T + (-29.7 - 8.73i)T^{2} \) |
| 37 | \( 1 + (7.67 - 2.25i)T + (31.1 - 20.0i)T^{2} \) |
| 41 | \( 1 + (-7.38 - 2.16i)T + (34.4 + 22.1i)T^{2} \) |
| 43 | \( 1 + (1.41 + 9.83i)T + (-41.2 + 12.1i)T^{2} \) |
| 47 | \( 1 - 2.37T + 47T^{2} \) |
| 53 | \( 1 + (-0.850 + 0.981i)T + (-7.54 - 52.4i)T^{2} \) |
| 59 | \( 1 + (-0.525 - 0.606i)T + (-8.39 + 58.3i)T^{2} \) |
| 61 | \( 1 + (-1.45 + 10.0i)T + (-58.5 - 17.1i)T^{2} \) |
| 67 | \( 1 + (4.11 + 2.64i)T + (27.8 + 60.9i)T^{2} \) |
| 71 | \( 1 + (-7.36 - 4.73i)T + (29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (-0.943 - 2.06i)T + (-47.8 + 55.1i)T^{2} \) |
| 79 | \( 1 + (6.49 + 7.49i)T + (-11.2 + 78.1i)T^{2} \) |
| 83 | \( 1 + (4.53 - 1.33i)T + (69.8 - 44.8i)T^{2} \) |
| 89 | \( 1 + (2.41 + 16.7i)T + (-85.3 + 25.0i)T^{2} \) |
| 97 | \( 1 + (-13.5 - 3.98i)T + (81.6 + 52.4i)T^{2} \) |
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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
−10.20966898261071046395198406566, −9.079387615504486556893850549702, −8.720898108317143799242232853724, −7.53692539363437907825641085516, −6.73627527654720574135029403125, −5.79908028435210396822719076971, −4.91555289223832469104566847833, −3.76308883742600742092137842865, −3.47418531446286795119561791222, −1.50179762009367961173683457489,
1.11878873647294198181760176268, 2.27844833251394801782022050620, 3.55296421568384211599132775646, 4.30540002193749248026504436269, 5.66878916466692334165790166226, 6.21499031255512304063018680522, 7.28969169759949755697065948536, 8.091459443301382173270325670546, 9.023600260685414669415927363990, 9.920120181837098199471113316986
