| L(s) = 1 | + (0.177 − 0.836i)3-s + (3.54 + 6.13i)5-s + (−3.95 + 1.76i)7-s + (7.55 + 3.36i)9-s + (3.23 + 0.340i)11-s + (2.24 + 2.02i)13-s + (5.76 − 1.87i)15-s + (16.4 − 1.73i)17-s + (−0.871 − 0.967i)19-s + (0.769 + 3.61i)21-s + (1.63 + 2.25i)23-s + (−12.6 + 21.8i)25-s + (8.67 − 11.9i)27-s + (−39.3 − 12.7i)29-s + (−30.7 − 4.25i)31-s + ⋯ |
| L(s) = 1 | + (0.0592 − 0.278i)3-s + (0.708 + 1.22i)5-s + (−0.564 + 0.251i)7-s + (0.839 + 0.373i)9-s + (0.294 + 0.0309i)11-s + (0.172 + 0.155i)13-s + (0.384 − 0.124i)15-s + (0.969 − 0.101i)17-s + (−0.0458 − 0.0509i)19-s + (0.0366 + 0.172i)21-s + (0.0711 + 0.0978i)23-s + (−0.504 + 0.873i)25-s + (0.321 − 0.442i)27-s + (−1.35 − 0.441i)29-s + (−0.990 − 0.137i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.795 - 0.605i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.795 - 0.605i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.48411 + 0.500801i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.48411 + 0.500801i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 31 | \( 1 + (30.7 + 4.25i)T \) |
| good | 3 | \( 1 + (-0.177 + 0.836i)T + (-8.22 - 3.66i)T^{2} \) |
| 5 | \( 1 + (-3.54 - 6.13i)T + (-12.5 + 21.6i)T^{2} \) |
| 7 | \( 1 + (3.95 - 1.76i)T + (32.7 - 36.4i)T^{2} \) |
| 11 | \( 1 + (-3.23 - 0.340i)T + (118. + 25.1i)T^{2} \) |
| 13 | \( 1 + (-2.24 - 2.02i)T + (17.6 + 168. i)T^{2} \) |
| 17 | \( 1 + (-16.4 + 1.73i)T + (282. - 60.0i)T^{2} \) |
| 19 | \( 1 + (0.871 + 0.967i)T + (-37.7 + 359. i)T^{2} \) |
| 23 | \( 1 + (-1.63 - 2.25i)T + (-163. + 503. i)T^{2} \) |
| 29 | \( 1 + (39.3 + 12.7i)T + (680. + 494. i)T^{2} \) |
| 37 | \( 1 + (16.1 + 9.35i)T + (684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-21.1 + 4.48i)T + (1.53e3 - 683. i)T^{2} \) |
| 43 | \( 1 + (-43.4 + 39.1i)T + (193. - 1.83e3i)T^{2} \) |
| 47 | \( 1 + (25.8 + 79.6i)T + (-1.78e3 + 1.29e3i)T^{2} \) |
| 53 | \( 1 + (-8.22 + 18.4i)T + (-1.87e3 - 2.08e3i)T^{2} \) |
| 59 | \( 1 + (82.8 + 17.6i)T + (3.18e3 + 1.41e3i)T^{2} \) |
| 61 | \( 1 - 12.8iT - 3.72e3T^{2} \) |
| 67 | \( 1 + (-42.9 - 74.4i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 + (9.33 + 4.15i)T + (3.37e3 + 3.74e3i)T^{2} \) |
| 73 | \( 1 + (-94.6 - 9.94i)T + (5.21e3 + 1.10e3i)T^{2} \) |
| 79 | \( 1 + (-104. + 10.9i)T + (6.10e3 - 1.29e3i)T^{2} \) |
| 83 | \( 1 + (-0.646 - 3.04i)T + (-6.29e3 + 2.80e3i)T^{2} \) |
| 89 | \( 1 + (-82.7 + 113. i)T + (-2.44e3 - 7.53e3i)T^{2} \) |
| 97 | \( 1 + (94.1 + 68.3i)T + (2.90e3 + 8.94e3i)T^{2} \) |
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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
−13.34587560418797675543335135903, −12.39012993913975829677059289520, −11.06351066036274627082192401074, −10.13641188770818741540442905620, −9.307888593169664089104263965399, −7.57630644840677255798042962630, −6.72737950155080540863355249118, −5.63024226485902058225714991808, −3.60519336954341073808629404459, −2.08225865542424167433318300842,
1.31689758334498411917969064990, 3.68375445104607823405011980527, 5.03887826255228431206364670083, 6.24370842997169410151075861563, 7.70459485318439362604728073699, 9.224372346755788138176265158655, 9.589661583996672897736179864722, 10.83555086017521990108125427268, 12.51803580005207419683558777069, 12.80031043779385997447936099376
