L(s) = 1 | + (−0.866 + 0.5i)3-s + (1.63 − 1.52i)5-s + (1.13 − 1.96i)7-s + (0.499 − 0.866i)9-s + (2.47 − 1.43i)11-s + (−1.36 − 3.33i)13-s + (−0.656 + 2.13i)15-s + (6.33 + 3.65i)17-s + (−6.64 − 3.83i)19-s + 2.26i·21-s + (−0.617 + 0.356i)23-s + (0.361 − 4.98i)25-s + 0.999i·27-s + (−3.35 − 5.81i)29-s + 2.56i·31-s + ⋯ |
L(s) = 1 | + (−0.499 + 0.288i)3-s + (0.732 − 0.681i)5-s + (0.428 − 0.741i)7-s + (0.166 − 0.288i)9-s + (0.746 − 0.431i)11-s + (−0.377 − 0.925i)13-s + (−0.169 + 0.551i)15-s + (1.53 + 0.886i)17-s + (−1.52 − 0.879i)19-s + 0.494i·21-s + (−0.128 + 0.0743i)23-s + (0.0722 − 0.997i)25-s + 0.192i·27-s + (−0.623 − 1.08i)29-s + 0.460i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0762 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0762 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.597459166\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.597459166\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.866 - 0.5i)T \) |
| 5 | \( 1 + (-1.63 + 1.52i)T \) |
| 13 | \( 1 + (1.36 + 3.33i)T \) |
good | 7 | \( 1 + (-1.13 + 1.96i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.47 + 1.43i)T + (5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (-6.33 - 3.65i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (6.64 + 3.83i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.617 - 0.356i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (3.35 + 5.81i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 2.56iT - 31T^{2} \) |
| 37 | \( 1 + (-3.00 - 5.20i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (4.15 - 2.40i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-1.62 - 0.935i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 9.49T + 47T^{2} \) |
| 53 | \( 1 + 8.14iT - 53T^{2} \) |
| 59 | \( 1 + (8.96 + 5.17i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.359 + 0.623i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.93 - 5.08i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (6.08 + 3.51i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 - 13.2T + 73T^{2} \) |
| 79 | \( 1 + 9.81T + 79T^{2} \) |
| 83 | \( 1 - 17.6T + 83T^{2} \) |
| 89 | \( 1 + (-13.0 + 7.54i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-5.88 + 10.1i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.410645035816230091379107171477, −8.376413337100977733656473647483, −7.85039599957201907905799587572, −6.55828660462723224091530516300, −5.98393034488636227274727648940, −5.07376957364194929054671514465, −4.36002713605096466350371286136, −3.33482069849346140618699440781, −1.76723598722929273002961171687, −0.68291933655768288385856496346,
1.59576748733528290194324537838, 2.30332522785117906744601980255, 3.64237213018143991608891716608, 4.82339023706590900806041150734, 5.66450428944345070437018914592, 6.34463723972029395764238163356, 7.09336857397194147897216056137, 7.88399793045223680853976833339, 9.047962967974907810942870738101, 9.570031505103962656136353762228