| L(s) = 1 | + (−1.68 + 0.357i)2-s + (−0.402 + 1.68i)3-s + (0.879 − 0.391i)4-s + (1.96 − 1.06i)5-s + (0.0746 − 2.98i)6-s + (−2.11 − 3.66i)7-s + (1.44 − 1.04i)8-s + (−2.67 − 1.35i)9-s + (−2.93 + 2.49i)10-s + (5.35 − 1.13i)11-s + (0.305 + 1.63i)12-s + (−2.27 − 0.483i)13-s + (4.87 + 5.41i)14-s + (0.996 + 3.74i)15-s + (−3.34 + 3.71i)16-s + (−1.13 + 0.824i)17-s + ⋯ |
| L(s) = 1 | + (−1.19 + 0.253i)2-s + (−0.232 + 0.972i)3-s + (0.439 − 0.195i)4-s + (0.880 − 0.474i)5-s + (0.0304 − 1.21i)6-s + (−0.799 − 1.38i)7-s + (0.510 − 0.370i)8-s + (−0.892 − 0.451i)9-s + (−0.927 + 0.788i)10-s + (1.61 − 0.343i)11-s + (0.0883 + 0.473i)12-s + (−0.630 − 0.133i)13-s + (1.30 + 1.44i)14-s + (0.257 + 0.966i)15-s + (−0.836 + 0.928i)16-s + (−0.275 + 0.199i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.978 + 0.204i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.978 + 0.204i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.628719 - 0.0648154i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.628719 - 0.0648154i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.402 - 1.68i)T \) |
| 5 | \( 1 + (-1.96 + 1.06i)T \) |
| good | 2 | \( 1 + (1.68 - 0.357i)T + (1.82 - 0.813i)T^{2} \) |
| 7 | \( 1 + (2.11 + 3.66i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-5.35 + 1.13i)T + (10.0 - 4.47i)T^{2} \) |
| 13 | \( 1 + (2.27 + 0.483i)T + (11.8 + 5.28i)T^{2} \) |
| 17 | \( 1 + (1.13 - 0.824i)T + (5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (-4.74 + 3.45i)T + (5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + (-2.33 - 2.59i)T + (-2.40 + 22.8i)T^{2} \) |
| 29 | \( 1 + (-0.451 + 4.29i)T + (-28.3 - 6.02i)T^{2} \) |
| 31 | \( 1 + (-0.330 - 3.14i)T + (-30.3 + 6.44i)T^{2} \) |
| 37 | \( 1 + (-0.930 - 2.86i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (4.52 + 0.961i)T + (37.4 + 16.6i)T^{2} \) |
| 43 | \( 1 + (2.17 + 3.76i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-0.300 + 2.85i)T + (-45.9 - 9.77i)T^{2} \) |
| 53 | \( 1 + (-4.61 - 3.35i)T + (16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (-11.3 - 2.41i)T + (53.8 + 23.9i)T^{2} \) |
| 61 | \( 1 + (10.6 - 2.25i)T + (55.7 - 24.8i)T^{2} \) |
| 67 | \( 1 + (1.17 + 11.2i)T + (-65.5 + 13.9i)T^{2} \) |
| 71 | \( 1 + (5.34 + 3.88i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-1.20 + 3.69i)T + (-59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (-0.299 + 2.85i)T + (-77.2 - 16.4i)T^{2} \) |
| 83 | \( 1 + (-6.23 - 2.77i)T + (55.5 + 61.6i)T^{2} \) |
| 89 | \( 1 + (1.41 - 4.35i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (1.44 - 13.7i)T + (-94.8 - 20.1i)T^{2} \) |
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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
−11.93150727241566778630547805984, −10.73797153809817474260485547231, −9.929795942261235917901921559029, −9.438685642717633060524098585469, −8.744327997967502172685458774148, −7.18951325922668238739163561028, −6.33217179457627226285001035503, −4.75950297809359884837891330074, −3.61246876459035501617388522566, −0.886822926074696129388715856988,
1.59288524023505184441269937439, 2.73034109251083173524428489238, 5.37257720639992648635184863834, 6.43106623031398250722649159474, 7.21681129253771133878753384201, 8.671548408060461552113700415031, 9.325793455597910143275614966830, 10.00745635742278103665039184680, 11.39952182749277942788125582022, 12.05078432540061452045059891924
