L(s) = 1 | + (−0.809 − 0.587i)2-s + (0.309 + 0.951i)4-s + (0.0348 + 0.0479i)5-s + (0.951 − 0.309i)7-s + (0.309 − 0.951i)8-s − 0.0592i·10-s + (0.764 − 3.22i)11-s + (−2.76 + 3.80i)13-s + (−0.951 − 0.309i)14-s + (−0.809 + 0.587i)16-s + (0.0206 − 0.0149i)17-s + (−5.33 − 1.73i)19-s + (−0.0348 + 0.0479i)20-s + (−2.51 + 2.16i)22-s − 7.57i·23-s + ⋯ |
L(s) = 1 | + (−0.572 − 0.415i)2-s + (0.154 + 0.475i)4-s + (0.0155 + 0.0214i)5-s + (0.359 − 0.116i)7-s + (0.109 − 0.336i)8-s − 0.0187i·10-s + (0.230 − 0.973i)11-s + (−0.765 + 1.05i)13-s + (−0.254 − 0.0825i)14-s + (−0.202 + 0.146i)16-s + (0.00499 − 0.00363i)17-s + (−1.22 − 0.397i)19-s + (−0.00779 + 0.0107i)20-s + (−0.536 + 0.460i)22-s − 1.57i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.567 + 0.823i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.567 + 0.823i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8466879833\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8466879833\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.809 + 0.587i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-0.951 + 0.309i)T \) |
| 11 | \( 1 + (-0.764 + 3.22i)T \) |
good | 5 | \( 1 + (-0.0348 - 0.0479i)T + (-1.54 + 4.75i)T^{2} \) |
| 13 | \( 1 + (2.76 - 3.80i)T + (-4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (-0.0206 + 0.0149i)T + (5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (5.33 + 1.73i)T + (15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + 7.57iT - 23T^{2} \) |
| 29 | \( 1 + (1.03 + 3.17i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-3.61 - 2.62i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (-1.34 - 4.14i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (0.884 - 2.72i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + 1.76iT - 43T^{2} \) |
| 47 | \( 1 + (6.78 + 2.20i)T + (38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (-5.47 + 7.53i)T + (-16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (-2.52 + 0.821i)T + (47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (-1.10 - 1.52i)T + (-18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + 6.35T + 67T^{2} \) |
| 71 | \( 1 + (8.20 + 11.2i)T + (-21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-0.975 + 0.316i)T + (59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (-7.60 + 10.4i)T + (-24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (-13.6 + 9.95i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 - 1.58iT - 89T^{2} \) |
| 97 | \( 1 + (13.5 + 9.81i)T + (29.9 + 92.2i)T^{2} \) |
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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
−9.194795997735966787883818700695, −8.556551853505489216991266104029, −7.987261447085762327186481450973, −6.72824558168503128627732598183, −6.36156805651041302057452586319, −4.82201036437942206787351027493, −4.17722809908582148919335929349, −2.86296249619845513245308548477, −1.94931969008815241108346855672, −0.42081855609270460125368702001,
1.41769927352209248508749940597, 2.54117430344791805679367795919, 3.93874671815685696095933815687, 5.04678398885629011988441524025, 5.69035759162504198405670920926, 6.80011983668374261913407255993, 7.53394804878213790305415663875, 8.105490461625468684692559693100, 9.113100071417796929985269661070, 9.728065488858708752987836222315