| 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
−9.920120181837098199471113316986, −9.023600260685414669415927363990, −8.091459443301382173270325670546, −7.28969169759949755697065948536, −6.21499031255512304063018680522, −5.66878916466692334165790166226, −4.30540002193749248026504436269, −3.55296421568384211599132775646, −2.27844833251394801782022050620, −1.11878873647294198181760176268,
1.50179762009367961173683457489, 3.47418531446286795119561791222, 3.76308883742600742092137842865, 4.91555289223832469104566847833, 5.79908028435210396822719076971, 6.73627527654720574135029403125, 7.53692539363437907825641085516, 8.720898108317143799242232853724, 9.079387615504486556893850549702, 10.20966898261071046395198406566
