L(s) = 1 | + (−2.11 + 0.567i)2-s + (−0.866 − 0.5i)3-s + (2.43 − 1.40i)4-s + (3.00 − 3.00i)5-s + (2.11 + 0.567i)6-s + (−2.49 − 0.883i)7-s + (−1.26 + 1.26i)8-s + (0.499 + 0.866i)9-s + (−4.65 + 8.06i)10-s + (−0.698 − 2.60i)11-s − 2.81·12-s + (−0.373 + 3.58i)13-s + (5.78 + 0.455i)14-s + (−4.10 + 1.09i)15-s + (−0.857 + 1.48i)16-s + (−0.599 − 1.03i)17-s + ⋯ |
L(s) = 1 | + (−1.49 + 0.401i)2-s + (−0.499 − 0.288i)3-s + (1.21 − 0.703i)4-s + (1.34 − 1.34i)5-s + (0.865 + 0.231i)6-s + (−0.942 − 0.333i)7-s + (−0.445 + 0.445i)8-s + (0.166 + 0.288i)9-s + (−1.47 + 2.55i)10-s + (−0.210 − 0.785i)11-s − 0.811·12-s + (−0.103 + 0.994i)13-s + (1.54 + 0.121i)14-s + (−1.05 + 0.283i)15-s + (−0.214 + 0.371i)16-s + (−0.145 − 0.251i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.406 + 0.913i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.406 + 0.913i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.238742 - 0.367476i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.238742 - 0.367476i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.866 + 0.5i)T \) |
| 7 | \( 1 + (2.49 + 0.883i)T \) |
| 13 | \( 1 + (0.373 - 3.58i)T \) |
good | 2 | \( 1 + (2.11 - 0.567i)T + (1.73 - i)T^{2} \) |
| 5 | \( 1 + (-3.00 + 3.00i)T - 5iT^{2} \) |
| 11 | \( 1 + (0.698 + 2.60i)T + (-9.52 + 5.5i)T^{2} \) |
| 17 | \( 1 + (0.599 + 1.03i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.89 + 0.507i)T + (16.4 + 9.5i)T^{2} \) |
| 23 | \( 1 + (4.65 + 2.68i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.47 + 2.56i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-3.36 + 3.36i)T - 31iT^{2} \) |
| 37 | \( 1 + (1.03 + 3.87i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (-1.42 - 5.32i)T + (-35.5 + 20.5i)T^{2} \) |
| 43 | \( 1 + (9.78 - 5.64i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-2.97 - 2.97i)T + 47iT^{2} \) |
| 53 | \( 1 + 11.0T + 53T^{2} \) |
| 59 | \( 1 + (-3.14 + 11.7i)T + (-51.0 - 29.5i)T^{2} \) |
| 61 | \( 1 + (-4.55 + 2.63i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-4.15 + 1.11i)T + (58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + (-0.800 + 2.98i)T + (-61.4 - 35.5i)T^{2} \) |
| 73 | \( 1 + (4.78 + 4.78i)T + 73iT^{2} \) |
| 79 | \( 1 + 1.16T + 79T^{2} \) |
| 83 | \( 1 + (-3.24 + 3.24i)T - 83iT^{2} \) |
| 89 | \( 1 + (4.80 - 1.28i)T + (77.0 - 44.5i)T^{2} \) |
| 97 | \( 1 + (-14.7 - 3.95i)T + (84.0 + 48.5i)T^{2} \) |
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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
−11.35320163691424629048463726330, −10.13510338239720581662206229247, −9.648472778466817630895997962340, −8.854613765251388660818291093053, −7.987118522726767482098830986232, −6.45343072531833120483190353788, −6.15493487640659658142308259768, −4.61264329937452663506006138771, −1.98330153234072908846588051536, −0.55393885334373411750177703138,
2.02369408505833311958241786730, 3.12247116358062291881509746428, 5.47615225991929069432483407395, 6.49357806892918095779836126705, 7.27625714837932237730073649941, 8.711179609800307054935415881993, 9.887116994271794491379804576938, 10.09271680895870271697064156554, 10.65575232601223084574023211806, 11.83821998430588818931820693305