L(s) = 1 | + (1.12 − 1.12i)2-s + (0.866 − 0.5i)3-s − 0.509i·4-s + (−0.973 + 3.63i)5-s + (0.409 − 1.53i)6-s + (2.28 − 1.33i)7-s + (1.66 + 1.66i)8-s + (0.499 − 0.866i)9-s + (2.97 + 5.15i)10-s + (0.872 − 3.25i)11-s + (−0.254 − 0.440i)12-s + (−3.03 − 1.94i)13-s + (1.06 − 4.05i)14-s + (0.973 + 3.63i)15-s + 4.75·16-s − 3.33·17-s + ⋯ |
L(s) = 1 | + (0.791 − 0.791i)2-s + (0.499 − 0.288i)3-s − 0.254i·4-s + (−0.435 + 1.62i)5-s + (0.167 − 0.624i)6-s + (0.863 − 0.504i)7-s + (0.590 + 0.590i)8-s + (0.166 − 0.288i)9-s + (0.942 + 1.63i)10-s + (0.263 − 0.982i)11-s + (−0.0734 − 0.127i)12-s + (−0.842 − 0.538i)13-s + (0.284 − 1.08i)14-s + (0.251 + 0.938i)15-s + 1.18·16-s − 0.809·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.888 + 0.457i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.888 + 0.457i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.11680 - 0.513167i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.11680 - 0.513167i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.866 + 0.5i)T \) |
| 7 | \( 1 + (-2.28 + 1.33i)T \) |
| 13 | \( 1 + (3.03 + 1.94i)T \) |
good | 2 | \( 1 + (-1.12 + 1.12i)T - 2iT^{2} \) |
| 5 | \( 1 + (0.973 - 3.63i)T + (-4.33 - 2.5i)T^{2} \) |
| 11 | \( 1 + (-0.872 + 3.25i)T + (-9.52 - 5.5i)T^{2} \) |
| 17 | \( 1 + 3.33T + 17T^{2} \) |
| 19 | \( 1 + (-0.733 + 0.196i)T + (16.4 - 9.5i)T^{2} \) |
| 23 | \( 1 - 3.56iT - 23T^{2} \) |
| 29 | \( 1 + (1.42 - 2.46i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (8.90 - 2.38i)T + (26.8 - 15.5i)T^{2} \) |
| 37 | \( 1 + (8.01 + 8.01i)T + 37iT^{2} \) |
| 41 | \( 1 + (-6.84 + 1.83i)T + (35.5 - 20.5i)T^{2} \) |
| 43 | \( 1 + (-10.9 + 6.31i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-1.32 - 0.356i)T + (40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (3.59 - 6.22i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (2.23 - 2.23i)T - 59iT^{2} \) |
| 61 | \( 1 + (-0.902 - 0.521i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.85 - 0.765i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + (7.00 + 1.87i)T + (61.4 + 35.5i)T^{2} \) |
| 73 | \( 1 + (-0.559 - 2.08i)T + (-63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (5.54 + 9.60i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (1.51 + 1.51i)T + 83iT^{2} \) |
| 89 | \( 1 + (2.23 - 2.23i)T - 89iT^{2} \) |
| 97 | \( 1 + (3.23 - 12.0i)T + (-84.0 - 48.5i)T^{2} \) |
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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
−11.74942029405809791911925487998, −10.82634933352467995572766651246, −10.71089493800684103387833686751, −8.927199283878914106882243223534, −7.54531634788305727051696124031, −7.29070450934222293795219989127, −5.58312811404146315314713784136, −4.02918781336817973062861246207, −3.25753202678447966573466552019, −2.17311345299113981405928074499,
1.82085414784283545296468566252, 4.35882243027021429091590363077, 4.58310325296008317472893513600, 5.57265012862513623076200842030, 7.12399803510518353135765806083, 8.019622621383167726787595064287, 9.001176708343027540479718219712, 9.728902834577808099609950148745, 11.26891951905095114470229897627, 12.41323763146052489199305131826