L(s) = 1 | + (0.951 + 0.309i)2-s + (0.587 + 0.809i)3-s + (0.809 + 0.587i)4-s + (−0.169 + 0.0550i)5-s + (0.309 + 0.951i)6-s + (2.60 − 0.462i)7-s + (0.587 + 0.809i)8-s + (−0.309 + 0.951i)9-s − 0.178·10-s + (1.87 − 2.73i)11-s + 0.999i·12-s + (−0.936 + 2.88i)13-s + (2.62 + 0.364i)14-s + (−0.144 − 0.104i)15-s + (0.309 + 0.951i)16-s + (−0.531 − 1.63i)17-s + ⋯ |
L(s) = 1 | + (0.672 + 0.218i)2-s + (0.339 + 0.467i)3-s + (0.404 + 0.293i)4-s + (−0.0757 + 0.0246i)5-s + (0.126 + 0.388i)6-s + (0.984 − 0.174i)7-s + (0.207 + 0.286i)8-s + (−0.103 + 0.317i)9-s − 0.0563·10-s + (0.564 − 0.825i)11-s + 0.288i·12-s + (−0.259 + 0.799i)13-s + (0.700 + 0.0975i)14-s + (−0.0372 − 0.0270i)15-s + (0.0772 + 0.237i)16-s + (−0.128 − 0.396i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.689 - 0.724i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.689 - 0.724i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.25880 + 0.968810i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.25880 + 0.968810i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.951 - 0.309i)T \) |
| 3 | \( 1 + (-0.587 - 0.809i)T \) |
| 7 | \( 1 + (-2.60 + 0.462i)T \) |
| 11 | \( 1 + (-1.87 + 2.73i)T \) |
good | 5 | \( 1 + (0.169 - 0.0550i)T + (4.04 - 2.93i)T^{2} \) |
| 13 | \( 1 + (0.936 - 2.88i)T + (-10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (0.531 + 1.63i)T + (-13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (2.99 - 2.17i)T + (5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 - 3.37T + 23T^{2} \) |
| 29 | \( 1 + (-1.15 + 1.58i)T + (-8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (-0.141 - 0.0460i)T + (25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (3.77 + 2.74i)T + (11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (-0.410 + 0.298i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 - 1.47iT - 43T^{2} \) |
| 47 | \( 1 + (5.74 + 7.90i)T + (-14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (-0.248 + 0.764i)T + (-42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (-0.274 + 0.377i)T + (-18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (4.45 + 13.7i)T + (-49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + 11.2T + 67T^{2} \) |
| 71 | \( 1 + (-0.790 - 2.43i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-7.70 - 5.59i)T + (22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (0.257 + 0.0837i)T + (63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (2.03 + 6.26i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + 17.2iT - 89T^{2} \) |
| 97 | \( 1 + (-4.38 - 1.42i)T + (78.4 + 57.0i)T^{2} \) |
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
−11.36348027649294476041801804581, −10.45126357812016311947821597978, −9.225445096333262535701517714176, −8.430810163986708949983380769226, −7.50052252882912351826291722614, −6.43794611075384983398028619979, −5.27667167367312418439185345407, −4.37141940490715900430379732432, −3.46025958157818424660591561427, −1.92400338477095224930753833339,
1.55707663839378534012947002763, 2.71536993692414711677154652763, 4.16077619099397483077501239223, 5.03906906293332729149279551402, 6.23089499023376662174122614609, 7.23866934194414002788097471081, 8.103659959519055707986683813023, 9.055213171782948385473823775571, 10.23469447205837715349018027697, 11.09755116766592255594708934542