L(s) = 1 | + (1.13 − 0.843i)2-s + (1.62 − 0.940i)3-s + (0.576 − 1.91i)4-s + (−1.68 + 1.47i)5-s + (1.05 − 2.44i)6-s + (−0.603 + 2.57i)7-s + (−0.960 − 2.66i)8-s + (0.267 − 0.463i)9-s + (−0.672 + 3.08i)10-s + (1.20 − 0.692i)11-s + (−0.861 − 3.66i)12-s − 5.79·13-s + (1.48 + 3.43i)14-s + (−1.36 + 3.97i)15-s + (−3.33 − 2.20i)16-s + (2.33 + 4.04i)17-s + ⋯ |
L(s) = 1 | + (0.802 − 0.596i)2-s + (0.940 − 0.542i)3-s + (0.288 − 0.957i)4-s + (−0.753 + 0.657i)5-s + (0.430 − 0.996i)6-s + (−0.228 + 0.973i)7-s + (−0.339 − 0.940i)8-s + (0.0891 − 0.154i)9-s + (−0.212 + 0.977i)10-s + (0.361 − 0.208i)11-s + (−0.248 − 1.05i)12-s − 1.60·13-s + (0.397 + 0.917i)14-s + (−0.351 + 1.02i)15-s + (−0.833 − 0.552i)16-s + (0.565 + 0.980i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.557 + 0.830i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.557 + 0.830i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.60239 - 0.854076i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.60239 - 0.854076i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.13 + 0.843i)T \) |
| 5 | \( 1 + (1.68 - 1.47i)T \) |
| 7 | \( 1 + (0.603 - 2.57i)T \) |
good | 3 | \( 1 + (-1.62 + 0.940i)T + (1.5 - 2.59i)T^{2} \) |
| 11 | \( 1 + (-1.20 + 0.692i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 5.79T + 13T^{2} \) |
| 17 | \( 1 + (-2.33 - 4.04i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.74 + 4.75i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.32 + 4.02i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 2.04T + 29T^{2} \) |
| 31 | \( 1 + (0.832 + 1.44i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (0.247 + 0.142i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 8.21iT - 41T^{2} \) |
| 43 | \( 1 + 3.30T + 43T^{2} \) |
| 47 | \( 1 + (0.119 + 0.0690i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-5.82 + 3.36i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-0.832 - 1.44i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (2.49 + 1.44i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (5.82 + 10.0i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 2.29iT - 71T^{2} \) |
| 73 | \( 1 + (2.33 + 4.04i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-8.88 - 5.12i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 0.0814iT - 83T^{2} \) |
| 89 | \( 1 + (3.22 + 1.85i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 3.88T + 97T^{2} \) |
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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
−12.89194885595829873671670047582, −12.16535226538556345272127029361, −11.29777654030366932146557546974, −10.01972200262566472551930571822, −8.847153438039891635056541399153, −7.58426913686068039163979996666, −6.49704965917582067094879477712, −4.91782664823083381107598201351, −3.22357072026904277251843295618, −2.41666487274403083838941794956,
3.18643146249361302528375746332, 4.12332186093855076112035059809, 5.20083492084338180342173321758, 7.19253700247045083855709420473, 7.74987449202283619113578143005, 9.032583838410371270101785799646, 9.991828738981388941924588119981, 11.75222203974096209729631470645, 12.36747376693842056951251025866, 13.63373505606641911870373989129