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
−11.35650316866934125919465634260, −10.76436802405053583424432967057, −9.308680256633918377232924860835, −8.368838322328773118248411384741, −7.66666530561339678887761819035, −6.40664297769111189135416655936, −6.03146038714164712206794579824, −4.77034883837876867742106296936, −3.36910768565193943223720006922, −2.81614083956952928509153933394,
0.69726467755071391428769687003, 2.38509880614900951322574353005, 3.90258830806155781236408016428, 4.47387986825853566929959941387, 5.55450139886772549197006332067, 6.76836362720351267160523670275, 7.71306230517162162343457063914, 8.980792944309677481240209905434, 9.855168737647367856767268330974, 10.59553963700138086422185828740