| L(s) = 1 | + (−0.164 − 1.14i)2-s + (0.415 − 0.909i)3-s + (0.636 − 0.186i)4-s + (−1.13 + 1.30i)5-s + (−1.10 − 0.325i)6-s + (−0.589 + 0.379i)7-s + (−1.27 − 2.80i)8-s + (−0.654 − 0.755i)9-s + (1.68 + 1.08i)10-s + (−0.485 + 3.37i)11-s + (0.0944 − 0.656i)12-s + (3.32 + 2.13i)13-s + (0.530 + 0.612i)14-s + (0.718 + 1.57i)15-s + (−1.87 + 1.20i)16-s + (0.920 + 0.270i)17-s + ⋯ |
| L(s) = 1 | + (−0.116 − 0.809i)2-s + (0.239 − 0.525i)3-s + (0.318 − 0.0934i)4-s + (−0.506 + 0.584i)5-s + (−0.452 − 0.132i)6-s + (−0.222 + 0.143i)7-s + (−0.452 − 0.990i)8-s + (−0.218 − 0.251i)9-s + (0.532 + 0.341i)10-s + (−0.146 + 1.01i)11-s + (0.0272 − 0.189i)12-s + (0.921 + 0.592i)13-s + (0.141 + 0.163i)14-s + (0.185 + 0.406i)15-s + (−0.469 + 0.301i)16-s + (0.223 + 0.0655i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.319 + 0.947i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.319 + 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.760629 - 0.546336i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.760629 - 0.546336i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.415 + 0.909i)T \) |
| 23 | \( 1 + (-2.27 + 4.22i)T \) |
| good | 2 | \( 1 + (0.164 + 1.14i)T + (-1.91 + 0.563i)T^{2} \) |
| 5 | \( 1 + (1.13 - 1.30i)T + (-0.711 - 4.94i)T^{2} \) |
| 7 | \( 1 + (0.589 - 0.379i)T + (2.90 - 6.36i)T^{2} \) |
| 11 | \( 1 + (0.485 - 3.37i)T + (-10.5 - 3.09i)T^{2} \) |
| 13 | \( 1 + (-3.32 - 2.13i)T + (5.40 + 11.8i)T^{2} \) |
| 17 | \( 1 + (-0.920 - 0.270i)T + (14.3 + 9.19i)T^{2} \) |
| 19 | \( 1 + (5.70 - 1.67i)T + (15.9 - 10.2i)T^{2} \) |
| 29 | \( 1 + (-2.25 - 0.661i)T + (24.3 + 15.6i)T^{2} \) |
| 31 | \( 1 + (3.58 + 7.85i)T + (-20.3 + 23.4i)T^{2} \) |
| 37 | \( 1 + (6.17 + 7.12i)T + (-5.26 + 36.6i)T^{2} \) |
| 41 | \( 1 + (4.08 - 4.71i)T + (-5.83 - 40.5i)T^{2} \) |
| 43 | \( 1 + (-3.32 + 7.27i)T + (-28.1 - 32.4i)T^{2} \) |
| 47 | \( 1 + 2.99T + 47T^{2} \) |
| 53 | \( 1 + (-3.96 + 2.54i)T + (22.0 - 48.2i)T^{2} \) |
| 59 | \( 1 + (-7.51 - 4.83i)T + (24.5 + 53.6i)T^{2} \) |
| 61 | \( 1 + (-3.64 - 7.97i)T + (-39.9 + 46.1i)T^{2} \) |
| 67 | \( 1 + (-0.241 - 1.67i)T + (-64.2 + 18.8i)T^{2} \) |
| 71 | \( 1 + (1.65 + 11.5i)T + (-68.1 + 20.0i)T^{2} \) |
| 73 | \( 1 + (-8.34 + 2.45i)T + (61.4 - 39.4i)T^{2} \) |
| 79 | \( 1 + (6.43 + 4.13i)T + (32.8 + 71.8i)T^{2} \) |
| 83 | \( 1 + (-9.33 - 10.7i)T + (-11.8 + 82.1i)T^{2} \) |
| 89 | \( 1 + (-4.45 + 9.74i)T + (-58.2 - 67.2i)T^{2} \) |
| 97 | \( 1 + (1.30 - 1.51i)T + (-13.8 - 96.0i)T^{2} \) |
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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
−14.65854434119570075295998473597, −13.08035278105645341599405720315, −12.26696361579018215275577058240, −11.21938801636549986282071331704, −10.30987359497568538877519234028, −8.904471464645944842557707348664, −7.31526482458421755486124778533, −6.34946477423079979463820768271, −3.78900843612807743279407843753, −2.16168548601745297890759846176,
3.38565853700966709840619747550, 5.25350465739549749246979845904, 6.61634161627239911988178186949, 8.215705594052768711476502949181, 8.706309333649692418570903581330, 10.54938790984446218984678434025, 11.52083597437003792299765905290, 12.91703211520890146822065318797, 14.14248144665601902318219879059, 15.37553505431722283168381007354
