| L(s) = 1 | + (0.342 + 0.939i)2-s + (−1.29 + 1.15i)3-s + (−0.766 + 0.642i)4-s + (−0.588 − 3.34i)5-s + (−1.52 − 0.819i)6-s + (−2.60 + 0.476i)7-s + (−0.866 − 0.500i)8-s + (0.338 − 2.98i)9-s + (2.93 − 1.69i)10-s + (5.82 + 1.02i)11-s + (0.248 − 1.71i)12-s + (1.71 − 4.72i)13-s + (−1.33 − 2.28i)14-s + (4.61 + 3.63i)15-s + (0.173 − 0.984i)16-s + (−0.893 − 1.54i)17-s + ⋯ |
| L(s) = 1 | + (0.241 + 0.664i)2-s + (−0.745 + 0.666i)3-s + (−0.383 + 0.321i)4-s + (−0.263 − 1.49i)5-s + (−0.622 − 0.334i)6-s + (−0.983 + 0.180i)7-s + (−0.306 − 0.176i)8-s + (0.112 − 0.993i)9-s + (0.928 − 0.536i)10-s + (1.75 + 0.309i)11-s + (0.0716 − 0.494i)12-s + (0.476 − 1.30i)13-s + (−0.357 − 0.610i)14-s + (1.19 + 0.938i)15-s + (0.0434 − 0.246i)16-s + (−0.216 − 0.375i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.885 + 0.464i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.885 + 0.464i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.869702 - 0.214176i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.869702 - 0.214176i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.342 - 0.939i)T \) |
| 3 | \( 1 + (1.29 - 1.15i)T \) |
| 7 | \( 1 + (2.60 - 0.476i)T \) |
| good | 5 | \( 1 + (0.588 + 3.34i)T + (-4.69 + 1.71i)T^{2} \) |
| 11 | \( 1 + (-5.82 - 1.02i)T + (10.3 + 3.76i)T^{2} \) |
| 13 | \( 1 + (-1.71 + 4.72i)T + (-9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (0.893 + 1.54i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.39 - 0.802i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (2.97 + 3.54i)T + (-3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (0.134 + 0.369i)T + (-22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (5.48 + 6.54i)T + (-5.38 + 30.5i)T^{2} \) |
| 37 | \( 1 + (3.02 + 5.24i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-4.59 - 1.67i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (-1.47 + 8.34i)T + (-40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (-4.18 - 3.50i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 - 3.16iT - 53T^{2} \) |
| 59 | \( 1 + (-0.911 - 5.17i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (2.75 - 3.28i)T + (-10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (10.1 + 3.68i)T + (51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (-8.84 + 5.10i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (1.36 + 0.786i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (12.3 - 4.47i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (4.62 - 1.68i)T + (63.5 - 53.3i)T^{2} \) |
| 89 | \( 1 + (0.381 - 0.661i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-6.77 - 1.19i)T + (91.1 + 33.1i)T^{2} \) |
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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
−11.53644746704329320098921782883, −10.18258200531603730315015408458, −9.164860552535056994702488252212, −8.879467193681773533062759771270, −7.41206383893585125860042712712, −6.14516664439256230619360732570, −5.59833821154377025885567636158, −4.37168545181310281149035423240, −3.70022281446550764848975495830, −0.65248249030016770788491860806,
1.65015945503170189685217791449, 3.25665674395994368905713064136, 4.12148878317001922921905115076, 6.00102487462233451081515694910, 6.61389672837324580088947171785, 7.20060936227458914932575973625, 8.893713303689148976396028200935, 9.878950009810589662573116289448, 10.85437741673991928172959391650, 11.48767374913604071494605913926
