| L(s) = 1 | + (0.959 − 0.281i)2-s + (−0.654 − 0.755i)3-s + (0.841 − 0.540i)4-s + (0.168 + 1.17i)5-s + (−0.841 − 0.540i)6-s + (−0.415 + 0.909i)7-s + (0.654 − 0.755i)8-s + (−0.142 + 0.989i)9-s + (0.491 + 1.07i)10-s + (−0.606 − 0.178i)11-s + (−0.959 − 0.281i)12-s + (0.134 + 0.295i)13-s + (−0.142 + 0.989i)14-s + (0.774 − 0.893i)15-s + (0.415 − 0.909i)16-s + (4.22 + 2.71i)17-s + ⋯ |
| L(s) = 1 | + (0.678 − 0.199i)2-s + (−0.378 − 0.436i)3-s + (0.420 − 0.270i)4-s + (0.0752 + 0.523i)5-s + (−0.343 − 0.220i)6-s + (−0.157 + 0.343i)7-s + (0.231 − 0.267i)8-s + (−0.0474 + 0.329i)9-s + (0.155 + 0.340i)10-s + (−0.182 − 0.0537i)11-s + (−0.276 − 0.0813i)12-s + (0.0374 + 0.0819i)13-s + (−0.0380 + 0.264i)14-s + (0.199 − 0.230i)15-s + (0.103 − 0.227i)16-s + (1.02 + 0.658i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 + 0.260i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.965 + 0.260i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.20769 - 0.293062i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.20769 - 0.293062i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.959 + 0.281i)T \) |
| 3 | \( 1 + (0.654 + 0.755i)T \) |
| 7 | \( 1 + (0.415 - 0.909i)T \) |
| 23 | \( 1 + (-4.26 + 2.18i)T \) |
| good | 5 | \( 1 + (-0.168 - 1.17i)T + (-4.79 + 1.40i)T^{2} \) |
| 11 | \( 1 + (0.606 + 0.178i)T + (9.25 + 5.94i)T^{2} \) |
| 13 | \( 1 + (-0.134 - 0.295i)T + (-8.51 + 9.82i)T^{2} \) |
| 17 | \( 1 + (-4.22 - 2.71i)T + (7.06 + 15.4i)T^{2} \) |
| 19 | \( 1 + (-6.35 + 4.08i)T + (7.89 - 17.2i)T^{2} \) |
| 29 | \( 1 + (-5.56 - 3.57i)T + (12.0 + 26.3i)T^{2} \) |
| 31 | \( 1 + (4.92 - 5.67i)T + (-4.41 - 30.6i)T^{2} \) |
| 37 | \( 1 + (0.424 - 2.95i)T + (-35.5 - 10.4i)T^{2} \) |
| 41 | \( 1 + (0.412 + 2.87i)T + (-39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (5.29 + 6.10i)T + (-6.11 + 42.5i)T^{2} \) |
| 47 | \( 1 + 6.34T + 47T^{2} \) |
| 53 | \( 1 + (-2.43 + 5.32i)T + (-34.7 - 40.0i)T^{2} \) |
| 59 | \( 1 + (-5.38 - 11.7i)T + (-38.6 + 44.5i)T^{2} \) |
| 61 | \( 1 + (6.79 - 7.83i)T + (-8.68 - 60.3i)T^{2} \) |
| 67 | \( 1 + (-11.7 + 3.44i)T + (56.3 - 36.2i)T^{2} \) |
| 71 | \( 1 + (-3.94 + 1.15i)T + (59.7 - 38.3i)T^{2} \) |
| 73 | \( 1 + (-8.15 + 5.24i)T + (30.3 - 66.4i)T^{2} \) |
| 79 | \( 1 + (5.68 + 12.4i)T + (-51.7 + 59.7i)T^{2} \) |
| 83 | \( 1 + (0.811 - 5.64i)T + (-79.6 - 23.3i)T^{2} \) |
| 89 | \( 1 + (2.49 + 2.87i)T + (-12.6 + 88.0i)T^{2} \) |
| 97 | \( 1 + (-2.77 - 19.2i)T + (-93.0 + 27.3i)T^{2} \) |
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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
−10.34801047104242466419399210940, −9.215689380929431798125919798520, −8.217544627596074627120396890713, −7.03582598759592572268510776836, −6.69095295169701054256243957207, −5.46978440994711716230191976204, −4.99521582742982673071849198755, −3.43297564561519950711232495897, −2.72250029374436739792468139608, −1.23563638843744206502702306129,
1.13193084375526616489113485970, 2.99535976695691403858008626236, 3.83351625143609113919048697457, 5.04783353510803407993200677614, 5.39165718908072359895549882321, 6.49446899468552585422144565368, 7.47316992392170449156528138211, 8.201099130947816498250569842309, 9.547208731552080066413890159666, 9.874721092987430971843338513152
