L(s) = 1 | + (1.02 − 0.969i)2-s + (0.118 − 1.99i)4-s + (−1.09 − 1.89i)5-s + (−1.40 − 2.24i)7-s + (−1.81 − 2.16i)8-s + (−2.95 − 0.887i)10-s + (−2.14 + 3.71i)11-s + 2.03·13-s + (−3.61 − 0.951i)14-s + (−3.97 − 0.472i)16-s + (−1.56 − 0.903i)17-s + (0.509 − 0.294i)19-s + (−3.90 + 1.95i)20-s + (1.39 + 5.90i)22-s + (−0.146 + 0.0846i)23-s + ⋯ |
L(s) = 1 | + (0.727 − 0.685i)2-s + (0.0591 − 0.998i)4-s + (−0.488 − 0.845i)5-s + (−0.529 − 0.848i)7-s + (−0.641 − 0.767i)8-s + (−0.935 − 0.280i)10-s + (−0.646 + 1.11i)11-s + 0.564·13-s + (−0.967 − 0.254i)14-s + (−0.992 − 0.118i)16-s + (−0.379 − 0.219i)17-s + (0.116 − 0.0674i)19-s + (−0.873 + 0.437i)20-s + (0.297 + 1.25i)22-s + (−0.0305 + 0.0176i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.944 + 0.327i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.944 + 0.327i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.247151 - 1.46617i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.247151 - 1.46617i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.02 + 0.969i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (1.40 + 2.24i)T \) |
good | 5 | \( 1 + (1.09 + 1.89i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (2.14 - 3.71i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 2.03T + 13T^{2} \) |
| 17 | \( 1 + (1.56 + 0.903i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.509 + 0.294i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.146 - 0.0846i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 0.439iT - 29T^{2} \) |
| 31 | \( 1 + (-2.82 + 4.88i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-7.72 + 4.46i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 10.6iT - 41T^{2} \) |
| 43 | \( 1 - 8.59T + 43T^{2} \) |
| 47 | \( 1 + (5.47 + 9.48i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-4.53 - 2.61i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-0.730 - 0.421i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (4.22 + 7.31i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (6.87 - 11.9i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 9.72iT - 71T^{2} \) |
| 73 | \( 1 + (-11.0 - 6.39i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (0.784 - 0.453i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 11.4iT - 83T^{2} \) |
| 89 | \( 1 + (-14.5 + 8.39i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 13.3iT - 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.59553963700138086422185828740, −9.855168737647367856767268330974, −8.980792944309677481240209905434, −7.71306230517162162343457063914, −6.76836362720351267160523670275, −5.55450139886772549197006332067, −4.47387986825853566929959941387, −3.90258830806155781236408016428, −2.38509880614900951322574353005, −0.69726467755071391428769687003,
2.81614083956952928509153933394, 3.36910768565193943223720006922, 4.77034883837876867742106296936, 6.03146038714164712206794579824, 6.40664297769111189135416655936, 7.66666530561339678887761819035, 8.368838322328773118248411384741, 9.308680256633918377232924860835, 10.76436802405053583424432967057, 11.35650316866934125919465634260