L(s) = 1 | + (−1.32 + 2.29i)3-s + (1.5 − 0.866i)5-s + (−2 − 3.46i)9-s + (−3.96 − 2.29i)11-s − 3.46i·13-s + 4.58i·15-s + (−4.5 − 2.59i)17-s + (−1.32 − 2.29i)19-s + (−3.96 + 2.29i)23-s + (−1 + 1.73i)25-s + 2.64·27-s + (1.32 − 2.29i)31-s + (10.5 − 6.06i)33-s + (−3.5 − 6.06i)37-s + (7.93 + 4.58i)39-s + ⋯ |
L(s) = 1 | + (−0.763 + 1.32i)3-s + (0.670 − 0.387i)5-s + (−0.666 − 1.15i)9-s + (−1.19 − 0.690i)11-s − 0.960i·13-s + 1.18i·15-s + (−1.09 − 0.630i)17-s + (−0.303 − 0.525i)19-s + (−0.827 + 0.477i)23-s + (−0.200 + 0.346i)25-s + 0.509·27-s + (0.237 − 0.411i)31-s + (1.82 − 1.05i)33-s + (−0.575 − 0.996i)37-s + (1.27 + 0.733i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0633 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0633 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.310339 - 0.330657i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.310339 - 0.330657i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (1.32 - 2.29i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (-1.5 + 0.866i)T + (2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (3.96 + 2.29i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 3.46iT - 13T^{2} \) |
| 17 | \( 1 + (4.5 + 2.59i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.32 + 2.29i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (3.96 - 2.29i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + (-1.32 + 2.29i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (3.5 + 6.06i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 3.46iT - 41T^{2} \) |
| 43 | \( 1 + 9.16iT - 43T^{2} \) |
| 47 | \( 1 + (-3.96 - 6.87i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (1.5 - 2.59i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.96 + 6.87i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (1.5 - 0.866i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.96 - 2.29i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 9.16iT - 71T^{2} \) |
| 73 | \( 1 + (-4.5 - 2.59i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (3.96 - 2.29i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + (1.5 - 0.866i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 3.46iT - 97T^{2} \) |
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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
−10.13760950578983016525979319809, −9.434859887994630105314078049790, −8.632761205257800317900545483038, −7.54972102471026283466427555756, −6.12802479011854783960056878058, −5.44286991589503904608586692744, −4.89774285576067412717145986331, −3.75517667811039268738461202582, −2.46950545956728809339955936577, −0.23108006771285037185584525750,
1.76393301916787636185043154828, 2.40879151630527325229761815194, 4.32123152525935443566052006544, 5.45913662093997898093211701397, 6.38679710477045818147983033576, 6.78061350528979505725629745195, 7.78061699448054457295937654883, 8.603253607937643016761616875479, 9.935071856214248231399385221297, 10.52325366973098981857599542714