L(s) = 1 | + (−0.443 + 1.34i)2-s + (0.0794 + 0.191i)3-s + (−1.60 − 1.19i)4-s + (0.707 + 0.292i)5-s + (−0.292 + 0.0215i)6-s + (−2.27 − 2.27i)7-s + (2.31 − 1.62i)8-s + (2.09 − 2.09i)9-s + (−0.707 + 0.819i)10-s + (−1.49 + 3.60i)11-s + (0.101 − 0.402i)12-s + (−4.50 + 1.86i)13-s + (4.05 − 2.04i)14-s + 0.158i·15-s + (1.15 + 3.82i)16-s + 3.05i·17-s + ⋯ |
L(s) = 1 | + (−0.313 + 0.949i)2-s + (0.0458 + 0.110i)3-s + (−0.803 − 0.595i)4-s + (0.316 + 0.130i)5-s + (−0.119 + 0.00880i)6-s + (−0.858 − 0.858i)7-s + (0.817 − 0.575i)8-s + (0.696 − 0.696i)9-s + (−0.223 + 0.259i)10-s + (−0.450 + 1.08i)11-s + (0.0291 − 0.116i)12-s + (−1.24 + 0.517i)13-s + (1.08 − 0.545i)14-s + 0.0410i·15-s + (0.289 + 0.957i)16-s + 0.740i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.576 - 0.817i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.576 - 0.817i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.534429 + 0.277037i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.534429 + 0.277037i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.443 - 1.34i)T \) |
good | 3 | \( 1 + (-0.0794 - 0.191i)T + (-2.12 + 2.12i)T^{2} \) |
| 5 | \( 1 + (-0.707 - 0.292i)T + (3.53 + 3.53i)T^{2} \) |
| 7 | \( 1 + (2.27 + 2.27i)T + 7iT^{2} \) |
| 11 | \( 1 + (1.49 - 3.60i)T + (-7.77 - 7.77i)T^{2} \) |
| 13 | \( 1 + (4.50 - 1.86i)T + (9.19 - 9.19i)T^{2} \) |
| 17 | \( 1 - 3.05iT - 17T^{2} \) |
| 19 | \( 1 + (-3.87 + 1.60i)T + (13.4 - 13.4i)T^{2} \) |
| 23 | \( 1 + (-0.271 + 0.271i)T - 23iT^{2} \) |
| 29 | \( 1 + (0.931 + 2.24i)T + (-20.5 + 20.5i)T^{2} \) |
| 31 | \( 1 - 6.82T + 31T^{2} \) |
| 37 | \( 1 + (-3.63 - 1.50i)T + (26.1 + 26.1i)T^{2} \) |
| 41 | \( 1 + (1.54 - 1.54i)T - 41iT^{2} \) |
| 43 | \( 1 + (-0.748 + 1.80i)T + (-30.4 - 30.4i)T^{2} \) |
| 47 | \( 1 - 7.37iT - 47T^{2} \) |
| 53 | \( 1 + (-1.67 + 4.04i)T + (-37.4 - 37.4i)T^{2} \) |
| 59 | \( 1 + (10.1 + 4.19i)T + (41.7 + 41.7i)T^{2} \) |
| 61 | \( 1 + (-1.35 - 3.28i)T + (-43.1 + 43.1i)T^{2} \) |
| 67 | \( 1 + (1.99 + 4.81i)T + (-47.3 + 47.3i)T^{2} \) |
| 71 | \( 1 + (-6.47 - 6.47i)T + 71iT^{2} \) |
| 73 | \( 1 + (2.84 - 2.84i)T - 73iT^{2} \) |
| 79 | \( 1 + 9.74iT - 79T^{2} \) |
| 83 | \( 1 + (9.04 - 3.74i)T + (58.6 - 58.6i)T^{2} \) |
| 89 | \( 1 + (-7.58 - 7.58i)T + 89iT^{2} \) |
| 97 | \( 1 - 3.71T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−17.09366087832283902287578884224, −15.87006652051185973190131434464, −14.92185428459844273959238902396, −13.66446509829547983844827284940, −12.50852223399705057322881970747, −10.02487162970451494527276710368, −9.674335322942448867537047349946, −7.51718548353460830738786687037, −6.53469146775854751516330082994, −4.45272389521002682795052947765,
2.82074860538497551180743379716, 5.30786678338673466944267923995, 7.72735097554307607334510753929, 9.305020759521921105361772097402, 10.25908696994540061538134632247, 11.82580475248531804649755308070, 12.91647944403095541967177358666, 13.81950652636914328114019056467, 15.74643805891306379893689329844, 16.82682638086308839166579305643