L(s) = 1 | + (−0.5 + 0.866i)2-s + (−0.5 − 0.866i)3-s + (−0.499 − 0.866i)4-s + (−0.227 + 0.393i)5-s + 0.999·6-s + (−0.227 + 2.63i)7-s + 0.999·8-s + (−0.499 + 0.866i)9-s + (−0.227 − 0.393i)10-s + (−0.5 − 0.866i)11-s + (−0.499 + 0.866i)12-s + 0.909·13-s + (−2.16 − 1.51i)14-s + 0.454·15-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)17-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.288 − 0.499i)3-s + (−0.249 − 0.433i)4-s + (−0.101 + 0.176i)5-s + 0.408·6-s + (−0.0859 + 0.996i)7-s + 0.353·8-s + (−0.166 + 0.288i)9-s + (−0.0719 − 0.124i)10-s + (−0.150 − 0.261i)11-s + (−0.144 + 0.249i)12-s + 0.252·13-s + (−0.579 − 0.404i)14-s + 0.117·15-s + (−0.125 + 0.216i)16-s + (−0.121 − 0.210i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.480 - 0.877i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.480 - 0.877i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.376584 + 0.635496i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.376584 + 0.635496i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.5 + 0.866i)T \) |
| 7 | \( 1 + (0.227 - 2.63i)T \) |
| 11 | \( 1 + (0.5 + 0.866i)T \) |
good | 5 | \( 1 + (0.227 - 0.393i)T + (-2.5 - 4.33i)T^{2} \) |
| 13 | \( 1 - 0.909T + 13T^{2} \) |
| 17 | \( 1 + (0.5 + 0.866i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.94 - 3.36i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (4.16 - 7.22i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 2.79T + 29T^{2} \) |
| 31 | \( 1 + (-4.79 - 8.30i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (3.94 - 6.82i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 1.79T + 41T^{2} \) |
| 43 | \( 1 - 7.88T + 43T^{2} \) |
| 47 | \( 1 + (0.169 - 0.292i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (0.454 + 0.787i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (3.48 + 6.03i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-7.02 + 12.1i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.89 - 3.28i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 4.79T + 71T^{2} \) |
| 73 | \( 1 + (5.33 + 9.24i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (2.68 - 4.64i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 1.97T + 83T^{2} \) |
| 89 | \( 1 + (-5.54 + 9.60i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 14.5T + 97T^{2} \) |
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
−11.39906557944883940293475129283, −10.39702717104053530987053348281, −9.379926216460516368183928720843, −8.498423017258260570315522193678, −7.73753851721326279164878502037, −6.67713450092300723062095075313, −5.87088351832905122411404056090, −5.05263811720458959983837046862, −3.33158883486157952311631093762, −1.70601479371730345511682395519,
0.53719408515018209833302588350, 2.46471176577586186006788628156, 4.02489999895463392542565645755, 4.52377027537023405314432433465, 6.06595842570451276506128523729, 7.16374600856770660544118281105, 8.231759369692251454082449355827, 9.093834079837554598264789420524, 10.16770966135559362284045564044, 10.58371292130581365900565632347