L(s) = 1 | + (−0.477 + 1.33i)2-s + (−1.45 + 0.934i)3-s + (−1.54 − 1.27i)4-s + (−0.5 − 0.866i)5-s + (−0.548 − 2.38i)6-s + (−1.98 − 1.14i)7-s + (2.42 − 1.44i)8-s + (1.25 − 2.72i)9-s + (1.39 − 0.251i)10-s + (4.17 + 2.40i)11-s + (3.43 + 0.410i)12-s + (−1.25 + 0.723i)13-s + (2.47 − 2.09i)14-s + (1.53 + 0.795i)15-s + (0.766 + 3.92i)16-s − 5.75i·17-s + ⋯ |
L(s) = 1 | + (−0.337 + 0.941i)2-s + (−0.841 + 0.539i)3-s + (−0.771 − 0.635i)4-s + (−0.223 − 0.387i)5-s + (−0.223 − 0.974i)6-s + (−0.750 − 0.433i)7-s + (0.859 − 0.511i)8-s + (0.417 − 0.908i)9-s + (0.440 − 0.0796i)10-s + (1.25 + 0.726i)11-s + (0.992 + 0.118i)12-s + (−0.347 + 0.200i)13-s + (0.661 − 0.559i)14-s + (0.397 + 0.205i)15-s + (0.191 + 0.981i)16-s − 1.39i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.897 - 0.440i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.897 - 0.440i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.686317 + 0.159200i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.686317 + 0.159200i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.477 - 1.33i)T \) |
| 3 | \( 1 + (1.45 - 0.934i)T \) |
| 5 | \( 1 + (0.5 + 0.866i)T \) |
good | 7 | \( 1 + (1.98 + 1.14i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-4.17 - 2.40i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (1.25 - 0.723i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + 5.75iT - 17T^{2} \) |
| 19 | \( 1 - 5.54T + 19T^{2} \) |
| 23 | \( 1 + (-0.878 - 1.52i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-3.85 + 6.67i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (2.38 - 1.37i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 1.48iT - 37T^{2} \) |
| 41 | \( 1 + (-5.84 + 3.37i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.513 + 0.889i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-5.87 + 10.1i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 14.2T + 53T^{2} \) |
| 59 | \( 1 + (7.06 - 4.08i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (2.27 + 1.31i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (0.258 + 0.448i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 8.60T + 71T^{2} \) |
| 73 | \( 1 - 1.59T + 73T^{2} \) |
| 79 | \( 1 + (7.04 + 4.06i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (7.73 + 4.46i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 7.90iT - 89T^{2} \) |
| 97 | \( 1 + (-1.55 + 2.69i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−11.61451818916758227867124278131, −10.20699166315989178229497793932, −9.570140041008068799431126834099, −9.030998412151582941993317133724, −7.32277349371609401967065261194, −6.88678238476333258949375159711, −5.71757196687199761470654963316, −4.73214039137364714629269842864, −3.82812310890723174323606512130, −0.76633084587998784279851842104,
1.18842091520894002173467851781, 2.89308051514211642108366516996, 4.07430338373875559315442817896, 5.58977995219136188871530417611, 6.56626877209132401160033054938, 7.64638439833619506637493808575, 8.771715176863881313891012535992, 9.704369672075102771139268348076, 10.72543161659764160324692418514, 11.32338186738736134007680612467