L(s) = 1 | + (0.0381 − 1.41i)2-s + (−1.59 + 0.679i)3-s + (−1.99 − 0.107i)4-s + (−0.642 + 2.39i)5-s + (0.899 + 2.27i)6-s + (−1.93 + 3.34i)7-s + (−0.228 + 2.81i)8-s + (2.07 − 2.16i)9-s + (3.36 + 0.999i)10-s + (−1.07 − 4.01i)11-s + (3.25 − 1.18i)12-s + (−0.850 + 3.17i)13-s + (4.65 + 2.85i)14-s + (−0.605 − 4.25i)15-s + (3.97 + 0.431i)16-s − 1.33i·17-s + ⋯ |
L(s) = 1 | + (0.0269 − 0.999i)2-s + (−0.919 + 0.392i)3-s + (−0.998 − 0.0539i)4-s + (−0.287 + 1.07i)5-s + (0.367 + 0.930i)6-s + (−0.730 + 1.26i)7-s + (−0.0809 + 0.996i)8-s + (0.692 − 0.721i)9-s + (1.06 + 0.316i)10-s + (−0.324 − 1.21i)11-s + (0.939 − 0.342i)12-s + (−0.235 + 0.880i)13-s + (1.24 + 0.764i)14-s + (−0.156 − 1.09i)15-s + (0.994 + 0.107i)16-s − 0.322i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.130 - 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.130 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.339117 + 0.297433i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.339117 + 0.297433i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.0381 + 1.41i)T \) |
| 3 | \( 1 + (1.59 - 0.679i)T \) |
good | 5 | \( 1 + (0.642 - 2.39i)T + (-4.33 - 2.5i)T^{2} \) |
| 7 | \( 1 + (1.93 - 3.34i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (1.07 + 4.01i)T + (-9.52 + 5.5i)T^{2} \) |
| 13 | \( 1 + (0.850 - 3.17i)T + (-11.2 - 6.5i)T^{2} \) |
| 17 | \( 1 + 1.33iT - 17T^{2} \) |
| 19 | \( 1 + (6.09 - 6.09i)T - 19iT^{2} \) |
| 23 | \( 1 + (-0.521 + 0.301i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.0730 - 0.272i)T + (-25.1 + 14.5i)T^{2} \) |
| 31 | \( 1 + (-5.84 + 3.37i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.00346 + 0.00346i)T - 37iT^{2} \) |
| 41 | \( 1 + (-0.614 - 1.06i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.563 + 0.151i)T + (37.2 - 21.5i)T^{2} \) |
| 47 | \( 1 + (1.24 - 2.16i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-3.24 - 3.24i)T + 53iT^{2} \) |
| 59 | \( 1 + (-5.40 - 1.44i)T + (51.0 + 29.5i)T^{2} \) |
| 61 | \( 1 + (1.97 - 0.528i)T + (52.8 - 30.5i)T^{2} \) |
| 67 | \( 1 + (-9.90 - 2.65i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 - 7.85iT - 71T^{2} \) |
| 73 | \( 1 - 7.41iT - 73T^{2} \) |
| 79 | \( 1 + (-0.839 - 0.484i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (0.639 - 0.171i)T + (71.8 - 41.5i)T^{2} \) |
| 89 | \( 1 + 11.2T + 89T^{2} \) |
| 97 | \( 1 + (3.24 - 5.62i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−12.93529377817401434088660337230, −12.03234859489511522552574870158, −11.34231851944988317775352100965, −10.51087517060481373310910481022, −9.592598291934690770368606708790, −8.448947658022756859724505362551, −6.48878865509660466074812503460, −5.62934025295506840141376478861, −3.97336578543698135047116349476, −2.69412008080596795960356067949,
0.49870655354588888757209775808, 4.32711448840205731722768136509, 5.02596284936108854588171314247, 6.54423386204526333580698523899, 7.28901081105858364977594860245, 8.365930641107169004997211391176, 9.813712025392111253179512310611, 10.63078944895978599210780602625, 12.41816657034811505073230055574, 12.88907718712813966864609333489