| 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
−9.874721092987430971843338513152, −9.547208731552080066413890159666, −8.201099130947816498250569842309, −7.47316992392170449156528138211, −6.49446899468552585422144565368, −5.39165718908072359895549882321, −5.04783353510803407993200677614, −3.83351625143609113919048697457, −2.99535976695691403858008626236, −1.13193084375526616489113485970,
1.23563638843744206502702306129, 2.72250029374436739792468139608, 3.43297564561519950711232495897, 4.99521582742982673071849198755, 5.46978440994711716230191976204, 6.69095295169701054256243957207, 7.03582598759592572268510776836, 8.217544627596074627120396890713, 9.215689380929431798125919798520, 10.34801047104242466419399210940
