L(s) = 1 | − 7-s + (−1.5 + 0.866i)13-s + 19-s + (0.5 + 0.866i)25-s + 1.73i·31-s + 1.73i·37-s + (−0.5 + 0.866i)43-s + (−0.5 − 0.866i)61-s + (−1.5 + 0.866i)67-s + (0.5 − 0.866i)73-s + (1.5 + 0.866i)79-s + (1.5 − 0.866i)91-s − 1.73i·103-s + ⋯ |
L(s) = 1 | − 7-s + (−1.5 + 0.866i)13-s + 19-s + (0.5 + 0.866i)25-s + 1.73i·31-s + 1.73i·37-s + (−0.5 + 0.866i)43-s + (−0.5 − 0.866i)61-s + (−1.5 + 0.866i)67-s + (0.5 − 0.866i)73-s + (1.5 + 0.866i)79-s + (1.5 − 0.866i)91-s − 1.73i·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0977 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0977 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7436150819\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7436150819\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 5 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 7 | \( 1 + T + T^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 - 1.73iT - T^{2} \) |
| 37 | \( 1 - 1.73iT - T^{2} \) |
| 41 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 73 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + (0.5 + 0.866i)T^{2} \) |
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
−9.441745388027253689956561372730, −8.537485418341376265690210624144, −7.53189804481116670919883617897, −6.91681211065318024499585023462, −6.34826758260039717661456176861, −5.17710922816259131839983574731, −4.67648375550538696526344388330, −3.38553863517960802531946724868, −2.83421409205258426244237504865, −1.49569383600972657544422535565,
0.46855008322279034826699165570, 2.29035475215703533085928477681, 3.01845737590230745737867451544, 3.96229600867309390172410672302, 4.99497728007132371328790400023, 5.71516146990152639360945575936, 6.52321417514903310232637302742, 7.43020527753381586090306463394, 7.81681491718672206284192146434, 8.995895180678542776277115577399