L(s) = 1 | + (1.35 − 0.403i)2-s + (−1.42 + 2.47i)3-s + (1.67 − 1.09i)4-s + (−0.139 + 0.242i)5-s + (−0.935 + 3.92i)6-s − 1.55i·7-s + (1.82 − 2.15i)8-s + (−2.57 − 4.45i)9-s + (−0.0917 + 0.385i)10-s − 2.44i·11-s + (0.317 + 5.69i)12-s + (−5.47 + 3.16i)13-s + (−0.627 − 2.10i)14-s + (−0.399 − 0.691i)15-s + (1.60 − 3.66i)16-s + (2.18 − 3.78i)17-s + ⋯ |
L(s) = 1 | + (0.958 − 0.285i)2-s + (−0.823 + 1.42i)3-s + (0.836 − 0.547i)4-s + (−0.0625 + 0.108i)5-s + (−0.382 + 1.60i)6-s − 0.586i·7-s + (0.645 − 0.763i)8-s + (−0.857 − 1.48i)9-s + (−0.0290 + 0.121i)10-s − 0.735i·11-s + (0.0916 + 1.64i)12-s + (−1.51 + 0.876i)13-s + (−0.167 − 0.562i)14-s + (−0.103 − 0.178i)15-s + (0.400 − 0.916i)16-s + (0.529 − 0.917i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.902 - 0.430i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.902 - 0.430i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.18271 + 0.267332i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.18271 + 0.267332i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.35 + 0.403i)T \) |
| 19 | \( 1 + (-3.17 - 2.99i)T \) |
good | 3 | \( 1 + (1.42 - 2.47i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (0.139 - 0.242i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + 1.55iT - 7T^{2} \) |
| 11 | \( 1 + 2.44iT - 11T^{2} \) |
| 13 | \( 1 + (5.47 - 3.16i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-2.18 + 3.78i)T + (-8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (5.71 - 3.30i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.695 + 0.401i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 1.42T + 31T^{2} \) |
| 37 | \( 1 - 4.74iT - 37T^{2} \) |
| 41 | \( 1 + (5.84 + 3.37i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (0.146 + 0.0845i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (1.19 - 0.691i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-4.25 + 2.45i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (1.05 - 1.81i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (1.58 + 2.73i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.00 - 1.73i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-1.74 + 3.02i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (5.43 - 9.41i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (3.00 - 5.19i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 9.99iT - 83T^{2} \) |
| 89 | \( 1 + (-7.76 + 4.48i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (1.77 + 1.02i)T + (48.5 + 84.0i)T^{2} \) |
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
−14.53957228589133853088491113428, −13.78451726080033103566549165998, −11.97974217022629416659456963391, −11.51024252707018815353350553237, −10.26091541110453733223911721316, −9.654242134399059339076241709172, −7.21843858331729021868962132983, −5.65959240621497333581773984732, −4.69477559169631948134796612158, −3.46696304127139788796125941931,
2.36775801665520762926267996118, 4.96623737727796089582688977149, 6.03157357361439118424871611415, 7.17199921744983239652552839350, 8.056225720746456316155797589670, 10.37002960980105190475046673100, 11.98736732265771807874944005105, 12.24756423969305952928374146001, 13.05219341102750161553230017546, 14.31688501965218580791360188964