L(s) = 1 | + (−0.5 − 0.866i)2-s + (0.866 + 0.5i)3-s + (−0.499 + 0.866i)4-s + (2.16 + 0.569i)5-s − 0.999i·6-s + (1.99 + 1.15i)7-s + 0.999·8-s + (0.499 + 0.866i)9-s + (−0.588 − 2.15i)10-s + 3.60·11-s + (−0.866 + 0.499i)12-s + (2.08 − 3.60i)13-s − 2.30i·14-s + (1.58 + 1.57i)15-s + (−0.5 − 0.866i)16-s + (−1.64 − 2.84i)17-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (0.499 + 0.288i)3-s + (−0.249 + 0.433i)4-s + (0.967 + 0.254i)5-s − 0.408i·6-s + (0.755 + 0.436i)7-s + 0.353·8-s + (0.166 + 0.288i)9-s + (−0.186 − 0.682i)10-s + 1.08·11-s + (−0.249 + 0.144i)12-s + (0.577 − 1.00i)13-s − 0.616i·14-s + (0.410 + 0.406i)15-s + (−0.125 − 0.216i)16-s + (−0.398 − 0.689i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 + 0.240i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.970 + 0.240i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.167709436\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.167709436\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 + 0.866i)T \) |
| 3 | \( 1 + (-0.866 - 0.5i)T \) |
| 5 | \( 1 + (-2.16 - 0.569i)T \) |
| 37 | \( 1 + (4.22 + 4.37i)T \) |
good | 7 | \( 1 + (-1.99 - 1.15i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 - 3.60T + 11T^{2} \) |
| 13 | \( 1 + (-2.08 + 3.60i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (1.64 + 2.84i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (3.53 + 2.04i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + 0.748T + 23T^{2} \) |
| 29 | \( 1 - 3.31iT - 29T^{2} \) |
| 31 | \( 1 - 4.54iT - 31T^{2} \) |
| 41 | \( 1 + (1.97 - 3.41i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 - 11.6T + 43T^{2} \) |
| 47 | \( 1 + 1.55iT - 47T^{2} \) |
| 53 | \( 1 + (2.02 - 1.16i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (10.1 - 5.86i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.15 - 1.24i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (6.87 + 3.97i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-0.361 + 0.626i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + 0.278iT - 73T^{2} \) |
| 79 | \( 1 + (-7.52 - 4.34i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (5.53 - 3.19i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (5.28 - 3.05i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 4.51T + 97T^{2} \) |
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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
−9.744305900148636160472479303736, −8.924807039752061482538454389596, −8.656037932947508902389637956542, −7.48235785514924824152638705028, −6.47554855832501699703517998521, −5.43447506408445673780384153290, −4.48666230496618655801605342552, −3.30688823510568111116209297251, −2.36503407596511908040978326330, −1.36131873826446378753001062745,
1.35112504949512791486924642288, 2.04200586120676781464726722790, 3.93282439281401869091879580007, 4.61909822874089026753251681695, 6.06869443590856024157499915266, 6.39667450577065159478751458089, 7.42636959488721080739094698117, 8.375841684088951932848177982827, 8.930541768289101663862049511970, 9.584887452458521699493842203924