L(s) = 1 | + (0.5 + 0.866i)2-s + (−0.866 − 0.5i)3-s + (−0.499 + 0.866i)4-s + (2.18 − 0.487i)5-s − 0.999i·6-s + (−2.13 − 1.23i)7-s − 0.999·8-s + (0.499 + 0.866i)9-s + (1.51 + 1.64i)10-s − 0.452·11-s + (0.866 − 0.499i)12-s + (1.54 − 2.67i)13-s − 2.46i·14-s + (−2.13 − 0.668i)15-s + (−0.5 − 0.866i)16-s + (−1.05 − 1.83i)17-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.499 − 0.288i)3-s + (−0.249 + 0.433i)4-s + (0.975 − 0.218i)5-s − 0.408i·6-s + (−0.806 − 0.465i)7-s − 0.353·8-s + (0.166 + 0.288i)9-s + (0.478 + 0.520i)10-s − 0.136·11-s + (0.249 − 0.144i)12-s + (0.428 − 0.741i)13-s − 0.658i·14-s + (−0.550 − 0.172i)15-s + (−0.125 − 0.216i)16-s + (−0.256 − 0.444i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.863 + 0.504i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.863 + 0.504i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.607296203\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.607296203\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 - 0.866i)T \) |
| 3 | \( 1 + (0.866 + 0.5i)T \) |
| 5 | \( 1 + (-2.18 + 0.487i)T \) |
| 37 | \( 1 + (0.0294 + 6.08i)T \) |
good | 7 | \( 1 + (2.13 + 1.23i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + 0.452T + 11T^{2} \) |
| 13 | \( 1 + (-1.54 + 2.67i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (1.05 + 1.83i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.58 - 1.49i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 - 1.77T + 23T^{2} \) |
| 29 | \( 1 + 8.36iT - 29T^{2} \) |
| 31 | \( 1 + 2.55iT - 31T^{2} \) |
| 41 | \( 1 + (1.18 - 2.04i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 - 3.94T + 43T^{2} \) |
| 47 | \( 1 - 2.64iT - 47T^{2} \) |
| 53 | \( 1 + (-8.26 + 4.77i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (2.78 - 1.60i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-7.10 - 4.10i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (8.26 + 4.77i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-0.00323 + 0.00560i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + 9.13iT - 73T^{2} \) |
| 79 | \( 1 + (-10.9 - 6.33i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (6.55 - 3.78i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-9.02 + 5.21i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 5.64T + 97T^{2} \) |
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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
−9.785387074080732304174705379391, −8.990448518391012232311448753883, −7.897550012339963096037475115419, −7.12075064110640963998197375360, −6.21144742564677217196215847649, −5.75177876018039112174410755700, −4.83910873288478030954429429127, −3.65894515001825754343460386653, −2.44531283826874575521477303590, −0.72174622357726519454714010221,
1.39910509283982171600110546583, 2.66696337073177961448449785044, 3.57192401211438711863689549632, 4.82109658293698062012793619016, 5.58679735686059192377817181177, 6.37551850140840997105301689297, 7.02504816213991117304910617233, 8.778644004784948638600037796864, 9.225098378606194144650359317021, 10.06503306134532425907057363477