| L(s) = 1 | + (−1.10 + 5.20i)3-s + (1.56 + 2.70i)5-s + (−1.90 + 0.848i)7-s + (−17.6 − 7.85i)9-s + (5.30 + 0.557i)11-s + (−2.91 − 2.62i)13-s + (−15.7 + 5.12i)15-s + (−26.6 + 2.79i)17-s + (14.0 + 15.6i)19-s + (−2.30 − 10.8i)21-s + (15.7 + 21.6i)23-s + (7.63 − 13.2i)25-s + (32.2 − 44.3i)27-s + (38.7 + 12.6i)29-s + (6.68 + 30.2i)31-s + ⋯ |
| L(s) = 1 | + (−0.368 + 1.73i)3-s + (0.312 + 0.540i)5-s + (−0.272 + 0.121i)7-s + (−1.95 − 0.872i)9-s + (0.482 + 0.0506i)11-s + (−0.224 − 0.202i)13-s + (−1.05 + 0.341i)15-s + (−1.56 + 0.164i)17-s + (0.740 + 0.822i)19-s + (−0.109 − 0.517i)21-s + (0.683 + 0.940i)23-s + (0.305 − 0.528i)25-s + (1.19 − 1.64i)27-s + (1.33 + 0.434i)29-s + (0.215 + 0.976i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.844 - 0.535i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.844 - 0.535i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.304488 + 1.04900i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.304488 + 1.04900i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 31 | \( 1 + (-6.68 - 30.2i)T \) |
| good | 3 | \( 1 + (1.10 - 5.20i)T + (-8.22 - 3.66i)T^{2} \) |
| 5 | \( 1 + (-1.56 - 2.70i)T + (-12.5 + 21.6i)T^{2} \) |
| 7 | \( 1 + (1.90 - 0.848i)T + (32.7 - 36.4i)T^{2} \) |
| 11 | \( 1 + (-5.30 - 0.557i)T + (118. + 25.1i)T^{2} \) |
| 13 | \( 1 + (2.91 + 2.62i)T + (17.6 + 168. i)T^{2} \) |
| 17 | \( 1 + (26.6 - 2.79i)T + (282. - 60.0i)T^{2} \) |
| 19 | \( 1 + (-14.0 - 15.6i)T + (-37.7 + 359. i)T^{2} \) |
| 23 | \( 1 + (-15.7 - 21.6i)T + (-163. + 503. i)T^{2} \) |
| 29 | \( 1 + (-38.7 - 12.6i)T + (680. + 494. i)T^{2} \) |
| 37 | \( 1 + (38.9 + 22.5i)T + (684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-75.9 + 16.1i)T + (1.53e3 - 683. i)T^{2} \) |
| 43 | \( 1 + (7.17 - 6.46i)T + (193. - 1.83e3i)T^{2} \) |
| 47 | \( 1 + (6.22 + 19.1i)T + (-1.78e3 + 1.29e3i)T^{2} \) |
| 53 | \( 1 + (11.3 - 25.4i)T + (-1.87e3 - 2.08e3i)T^{2} \) |
| 59 | \( 1 + (-67.6 - 14.3i)T + (3.18e3 + 1.41e3i)T^{2} \) |
| 61 | \( 1 - 32.5iT - 3.72e3T^{2} \) |
| 67 | \( 1 + (13.0 + 22.5i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 + (-40.1 - 17.8i)T + (3.37e3 + 3.74e3i)T^{2} \) |
| 73 | \( 1 + (-109. - 11.5i)T + (5.21e3 + 1.10e3i)T^{2} \) |
| 79 | \( 1 + (110. - 11.6i)T + (6.10e3 - 1.29e3i)T^{2} \) |
| 83 | \( 1 + (12.7 + 59.8i)T + (-6.29e3 + 2.80e3i)T^{2} \) |
| 89 | \( 1 + (-16.5 + 22.8i)T + (-2.44e3 - 7.53e3i)T^{2} \) |
| 97 | \( 1 + (-41.0 - 29.8i)T + (2.90e3 + 8.94e3i)T^{2} \) |
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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
−13.94755478361546024235546701902, −12.33271095020242437219726300845, −11.17034261733389221661008975481, −10.46408092394949060088409173909, −9.580960568736651149168392527461, −8.716134373886410684650403376683, −6.75406768471845004414538253563, −5.54728935997492331864541928307, −4.35096522789609633898525736868, −3.06856428916587107711160292922,
0.815996006868264073285144165241, 2.43301790971595540149692268392, 4.90820829574646170947236728827, 6.39867766766888514463043557728, 6.98611977217118607099543220065, 8.328284286496228447301696649624, 9.328179748717690627981610660947, 11.08600187866514570556820463030, 11.88806011179206346237635860072, 12.93981612524957338697824775746
