L(s) = 1 | + (−1.35 + 0.402i)2-s + (1.67 − 1.09i)4-s + (2.11 − 1.22i)5-s + (−1.64 + 2.06i)7-s + (−1.83 + 2.15i)8-s + (−2.38 + 2.51i)10-s + (−5.38 − 3.11i)11-s + 2.57i·13-s + (1.40 − 3.46i)14-s + (1.61 − 3.65i)16-s + (−5.44 − 3.14i)17-s + (−2.28 − 3.96i)19-s + (2.21 − 4.36i)20-s + (8.55 + 2.04i)22-s + (1.66 − 0.963i)23-s + ⋯ |
L(s) = 1 | + (−0.958 + 0.284i)2-s + (0.837 − 0.545i)4-s + (0.947 − 0.547i)5-s + (−0.622 + 0.782i)7-s + (−0.647 + 0.761i)8-s + (−0.752 + 0.794i)10-s + (−1.62 − 0.938i)11-s + 0.713i·13-s + (0.374 − 0.927i)14-s + (0.403 − 0.914i)16-s + (−1.32 − 0.762i)17-s + (−0.525 − 0.909i)19-s + (0.495 − 0.976i)20-s + (1.82 + 0.436i)22-s + (0.348 − 0.200i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.849 + 0.527i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.849 + 0.527i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0620794 - 0.217765i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0620794 - 0.217765i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.35 - 0.402i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (1.64 - 2.06i)T \) |
good | 5 | \( 1 + (-2.11 + 1.22i)T + (2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (5.38 + 3.11i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 2.57iT - 13T^{2} \) |
| 17 | \( 1 + (5.44 + 3.14i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (2.28 + 3.96i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.66 + 0.963i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 6.67T + 29T^{2} \) |
| 31 | \( 1 + (1.50 - 2.59i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.60 - 2.77i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 2.33iT - 41T^{2} \) |
| 43 | \( 1 + 5.77iT - 43T^{2} \) |
| 47 | \( 1 + (-4.53 - 7.85i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-2.91 + 5.04i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.10 + 1.91i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (8.57 - 4.95i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (13.4 + 7.73i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 0.552iT - 71T^{2} \) |
| 73 | \( 1 + (-6.82 - 3.93i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-10.7 + 6.20i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 0.324T + 83T^{2} \) |
| 89 | \( 1 + (8.69 - 5.02i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 17.2iT - 97T^{2} \) |
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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
−9.721410058141872610329687792760, −8.974524788077047645082433462323, −8.727965572192188619589673676544, −7.43439629987996425414214260806, −6.48057275171162259143204754905, −5.68508287933274558368183490374, −4.96862261386493585638117733973, −2.82553610738029520403981338487, −2.07573887866032698816087436770, −0.13808188888289643512169260785,
1.92834930406166167515144880423, 2.75886368362648249849160450133, 4.05782157583201904721466017520, 5.66221136885089023450177431530, 6.49454662084585765719575395300, 7.40084196003247356280669024707, 8.036123449624297918229176502079, 9.268159246447145550727918124881, 9.975568434357339266144597033235, 10.61195804622347654038487205117