L(s) = 1 | + (0.5 + 0.866i)3-s + (0.5 − 0.866i)5-s + (0.5 + 0.866i)7-s + (−0.499 + 0.866i)9-s + (0.5 + 0.866i)11-s + (1 − 1.73i)13-s + 0.999·15-s − 2·17-s + (−0.499 + 0.866i)21-s + (−0.499 − 0.866i)25-s − 0.999·27-s + (0.5 + 0.866i)29-s + (−0.499 + 0.866i)33-s + 0.999·35-s + 1.99·39-s + ⋯ |
L(s) = 1 | + (0.5 + 0.866i)3-s + (0.5 − 0.866i)5-s + (0.5 + 0.866i)7-s + (−0.499 + 0.866i)9-s + (0.5 + 0.866i)11-s + (1 − 1.73i)13-s + 0.999·15-s − 2·17-s + (−0.499 + 0.866i)21-s + (−0.499 − 0.866i)25-s − 0.999·27-s + (0.5 + 0.866i)29-s + (−0.499 + 0.866i)33-s + 0.999·35-s + 1.99·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.416075455\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.416075455\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.5 - 0.866i)T \) |
| 5 | \( 1 + (-0.5 + 0.866i)T \) |
| 7 | \( 1 + (-0.5 - 0.866i)T \) |
good | 11 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 + 2T + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + T + T^{2} \) |
| 73 | \( 1 - T + T^{2} \) |
| 79 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
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
−9.864469230673860782935480899690, −8.950824591696030781574966156210, −8.668079352101774891437031947705, −7.927287338321563240811954957928, −6.51042018249083745981174086560, −5.50908495733405402775612947779, −4.90388472246914989302735137791, −4.06775105549190429450854116228, −2.78001625726864448268340408729, −1.75699795086470092166979045947,
1.45994906823908556845935474004, 2.36820505573428217124857433547, 3.63311597893621689150997821260, 4.40002034353953544830578766545, 6.13313138525700253975617482751, 6.55572790767979723827497821706, 7.12597916282249765740688335300, 8.220780031561802450934115196483, 8.901788031170369103447288862333, 9.594761034038477651683029524446