| L(s) = 1 | + (−1.03 − 1.35i)2-s + (−0.776 + 1.54i)3-s + (−0.233 + 0.871i)4-s + (−0.974 − 0.0638i)5-s + (2.89 − 0.556i)6-s + (−0.213 − 3.25i)7-s + (−1.72 + 0.715i)8-s + (−1.79 − 2.40i)9-s + (0.923 + 1.38i)10-s + (−2.56 − 2.24i)11-s + (−1.16 − 1.03i)12-s + (−1.81 − 0.485i)13-s + (−4.17 + 3.66i)14-s + (0.854 − 1.45i)15-s + (4.32 + 2.49i)16-s + (−3.35 − 2.39i)17-s + ⋯ |
| L(s) = 1 | + (−0.733 − 0.955i)2-s + (−0.448 + 0.893i)3-s + (−0.116 + 0.435i)4-s + (−0.435 − 0.0285i)5-s + (1.18 − 0.227i)6-s + (−0.0805 − 1.22i)7-s + (−0.611 + 0.253i)8-s + (−0.598 − 0.801i)9-s + (0.292 + 0.437i)10-s + (−0.772 − 0.677i)11-s + (−0.337 − 0.299i)12-s + (−0.502 − 0.134i)13-s + (−1.11 + 0.978i)14-s + (0.220 − 0.376i)15-s + (1.08 + 0.623i)16-s + (−0.813 − 0.581i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.969 + 0.245i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.969 + 0.245i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0411127 - 0.329265i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0411127 - 0.329265i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.776 - 1.54i)T \) |
| 17 | \( 1 + (3.35 + 2.39i)T \) |
| good | 2 | \( 1 + (1.03 + 1.35i)T + (-0.517 + 1.93i)T^{2} \) |
| 5 | \( 1 + (0.974 + 0.0638i)T + (4.95 + 0.652i)T^{2} \) |
| 7 | \( 1 + (0.213 + 3.25i)T + (-6.94 + 0.913i)T^{2} \) |
| 11 | \( 1 + (2.56 + 2.24i)T + (1.43 + 10.9i)T^{2} \) |
| 13 | \( 1 + (1.81 + 0.485i)T + (11.2 + 6.5i)T^{2} \) |
| 19 | \( 1 + (-1.65 - 0.683i)T + (13.4 + 13.4i)T^{2} \) |
| 23 | \( 1 + (-1.00 - 2.95i)T + (-18.2 + 14.0i)T^{2} \) |
| 29 | \( 1 + (-5.86 + 2.89i)T + (17.6 - 23.0i)T^{2} \) |
| 31 | \( 1 + (-0.750 - 0.855i)T + (-4.04 + 30.7i)T^{2} \) |
| 37 | \( 1 + (1.75 + 8.83i)T + (-34.1 + 14.1i)T^{2} \) |
| 41 | \( 1 + (-2.13 + 4.32i)T + (-24.9 - 32.5i)T^{2} \) |
| 43 | \( 1 + (1.45 - 11.0i)T + (-41.5 - 11.1i)T^{2} \) |
| 47 | \( 1 + (-3.51 + 0.942i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (0.730 - 1.76i)T + (-37.4 - 37.4i)T^{2} \) |
| 59 | \( 1 + (-5.03 - 3.86i)T + (15.2 + 56.9i)T^{2} \) |
| 61 | \( 1 + (-6.66 + 0.436i)T + (60.4 - 7.96i)T^{2} \) |
| 67 | \( 1 + (4.41 - 2.54i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (16.3 - 3.25i)T + (65.5 - 27.1i)T^{2} \) |
| 73 | \( 1 + (13.5 + 9.05i)T + (27.9 + 67.4i)T^{2} \) |
| 79 | \( 1 + (0.733 - 0.836i)T + (-10.3 - 78.3i)T^{2} \) |
| 83 | \( 1 + (-6.72 + 5.16i)T + (21.4 - 80.1i)T^{2} \) |
| 89 | \( 1 + (-9.18 + 9.18i)T - 89iT^{2} \) |
| 97 | \( 1 + (5.86 + 11.8i)T + (-59.0 + 76.9i)T^{2} \) |
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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
−11.97224892980657397660376787822, −11.23311002684575925800332143696, −10.49031570686808575823936397089, −9.839620448586801476686641123788, −8.750822228274600671398761869687, −7.45376185573306415122547394805, −5.79129839922262817970621718534, −4.34122209778701629557595814469, −3.03234761039981153133712264802, −0.40993340651369613418276598732,
2.53749229053572267481620071795, 5.10911483417149138756391318060, 6.27155440649544057501899513540, 7.16570194937670531019801394926, 8.131716104811642038911681834315, 8.868106133959467512059094056498, 10.25103995825012291606524093488, 11.77809482278278203840169125407, 12.28157388152539450540217936982, 13.26051353653407995303246299255
