L(s) = 1 | + (−0.892 − 0.239i)2-s + (−0.988 + 0.570i)3-s + (−0.993 − 0.573i)4-s + (2.80 + 2.80i)5-s + (1.01 − 0.272i)6-s + (2.62 + 0.297i)7-s + (2.05 + 2.05i)8-s + (−0.848 + 1.47i)9-s + (−1.82 − 3.16i)10-s + (0.544 − 2.03i)11-s + 1.30·12-s + (−3.41 + 1.16i)13-s + (−2.27 − 0.893i)14-s + (−4.36 − 1.16i)15-s + (−0.195 − 0.339i)16-s + (1.58 − 2.74i)17-s + ⋯ |
L(s) = 1 | + (−0.630 − 0.169i)2-s + (−0.570 + 0.329i)3-s + (−0.496 − 0.286i)4-s + (1.25 + 1.25i)5-s + (0.415 − 0.111i)6-s + (0.993 + 0.112i)7-s + (0.726 + 0.726i)8-s + (−0.282 + 0.490i)9-s + (−0.578 − 1.00i)10-s + (0.164 − 0.613i)11-s + 0.377·12-s + (−0.946 + 0.324i)13-s + (−0.607 − 0.238i)14-s + (−1.12 − 0.301i)15-s + (−0.0489 − 0.0847i)16-s + (0.384 − 0.666i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.754 - 0.655i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.754 - 0.655i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.641019 + 0.239553i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.641019 + 0.239553i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (-2.62 - 0.297i)T \) |
| 13 | \( 1 + (3.41 - 1.16i)T \) |
good | 2 | \( 1 + (0.892 + 0.239i)T + (1.73 + i)T^{2} \) |
| 3 | \( 1 + (0.988 - 0.570i)T + (1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (-2.80 - 2.80i)T + 5iT^{2} \) |
| 11 | \( 1 + (-0.544 + 2.03i)T + (-9.52 - 5.5i)T^{2} \) |
| 17 | \( 1 + (-1.58 + 2.74i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.12 - 0.302i)T + (16.4 - 9.5i)T^{2} \) |
| 23 | \( 1 + (-3.26 + 1.88i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (0.584 + 1.01i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (3.57 + 3.57i)T + 31iT^{2} \) |
| 37 | \( 1 + (-1.14 + 4.26i)T + (-32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + (1.85 - 6.93i)T + (-35.5 - 20.5i)T^{2} \) |
| 43 | \( 1 + (1.91 + 1.10i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-8.21 + 8.21i)T - 47iT^{2} \) |
| 53 | \( 1 - 4.89T + 53T^{2} \) |
| 59 | \( 1 + (0.0633 + 0.236i)T + (-51.0 + 29.5i)T^{2} \) |
| 61 | \( 1 + (10.7 + 6.18i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (9.95 + 2.66i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + (-1.60 - 5.98i)T + (-61.4 + 35.5i)T^{2} \) |
| 73 | \( 1 + (-3.47 + 3.47i)T - 73iT^{2} \) |
| 79 | \( 1 + 2.69T + 79T^{2} \) |
| 83 | \( 1 + (3.31 + 3.31i)T + 83iT^{2} \) |
| 89 | \( 1 + (6.71 + 1.80i)T + (77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 + (-15.8 + 4.24i)T + (84.0 - 48.5i)T^{2} \) |
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
−14.24916775943145943413506850366, −13.54151324218207218275275396467, −11.52073028872760935522511260766, −10.77776616364038767083511340484, −10.05572439053290747933909776472, −8.988756297658373773029505586362, −7.49963341096895321073233183208, −5.86417136039678683494191037365, −4.93770424340705836824871711781, −2.25587549835394130405221185605,
1.35608882081756549461653719191, 4.65124037445468927538801378503, 5.61564887071493962406462181427, 7.30407404201408821455840731754, 8.624727926558304107460929257108, 9.366382457566487579392667083362, 10.45762430039839212969504876029, 12.18592801246934344431270310372, 12.72730737591674703910915492735, 13.81615230339558488923477401182