L(s) = 1 | + (0.923 + 0.382i)3-s + (0.705 + 1.70i)5-s + (−3.24 + 3.24i)7-s + (0.707 + 0.707i)9-s + (−3.38 + 1.40i)11-s + (−0.503 + 1.21i)13-s + 1.84i·15-s + 0.622i·17-s + (2.14 − 5.17i)19-s + (−4.23 + 1.75i)21-s + (2.47 + 2.47i)23-s + (1.13 − 1.13i)25-s + (0.382 + 0.923i)27-s + (−2.16 − 0.897i)29-s + 10.4·31-s + ⋯ |
L(s) = 1 | + (0.533 + 0.220i)3-s + (0.315 + 0.762i)5-s + (−1.22 + 1.22i)7-s + (0.235 + 0.235i)9-s + (−1.02 + 0.423i)11-s + (−0.139 + 0.337i)13-s + 0.476i·15-s + 0.151i·17-s + (0.491 − 1.18i)19-s + (−0.924 + 0.382i)21-s + (0.516 + 0.516i)23-s + (0.226 − 0.226i)25-s + (0.0736 + 0.177i)27-s + (−0.402 − 0.166i)29-s + 1.87·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.147 - 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.147 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.871315 + 1.01138i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.871315 + 1.01138i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.923 - 0.382i)T \) |
good | 5 | \( 1 + (-0.705 - 1.70i)T + (-3.53 + 3.53i)T^{2} \) |
| 7 | \( 1 + (3.24 - 3.24i)T - 7iT^{2} \) |
| 11 | \( 1 + (3.38 - 1.40i)T + (7.77 - 7.77i)T^{2} \) |
| 13 | \( 1 + (0.503 - 1.21i)T + (-9.19 - 9.19i)T^{2} \) |
| 17 | \( 1 - 0.622iT - 17T^{2} \) |
| 19 | \( 1 + (-2.14 + 5.17i)T + (-13.4 - 13.4i)T^{2} \) |
| 23 | \( 1 + (-2.47 - 2.47i)T + 23iT^{2} \) |
| 29 | \( 1 + (2.16 + 0.897i)T + (20.5 + 20.5i)T^{2} \) |
| 31 | \( 1 - 10.4T + 31T^{2} \) |
| 37 | \( 1 + (0.0714 + 0.172i)T + (-26.1 + 26.1i)T^{2} \) |
| 41 | \( 1 + (-8.50 - 8.50i)T + 41iT^{2} \) |
| 43 | \( 1 + (-3.62 + 1.50i)T + (30.4 - 30.4i)T^{2} \) |
| 47 | \( 1 - 5.02iT - 47T^{2} \) |
| 53 | \( 1 + (7.15 - 2.96i)T + (37.4 - 37.4i)T^{2} \) |
| 59 | \( 1 + (-1.52 - 3.68i)T + (-41.7 + 41.7i)T^{2} \) |
| 61 | \( 1 + (3.07 + 1.27i)T + (43.1 + 43.1i)T^{2} \) |
| 67 | \( 1 + (-2.17 - 0.901i)T + (47.3 + 47.3i)T^{2} \) |
| 71 | \( 1 + (-1.11 + 1.11i)T - 71iT^{2} \) |
| 73 | \( 1 + (-3.71 - 3.71i)T + 73iT^{2} \) |
| 79 | \( 1 + 10.2iT - 79T^{2} \) |
| 83 | \( 1 + (-4.69 + 11.3i)T + (-58.6 - 58.6i)T^{2} \) |
| 89 | \( 1 + (3.54 - 3.54i)T - 89iT^{2} \) |
| 97 | \( 1 - 0.139T + 97T^{2} \) |
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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
−11.55114687142120845930056560332, −10.49425283539525992712929484642, −9.622062345481851470587914285592, −9.124270392046542124053217431035, −7.88088633498203363064705230440, −6.80493951420777498398618085261, −5.94385757870580015846784221202, −4.71728775616819799348881466963, −2.94483986303885731203335079788, −2.59720542308778751777539657441,
0.842993798610073370112353680567, 2.81428544617093241523203010771, 3.88217841139238467046962863358, 5.22320042651549348122842539848, 6.38645014400651757707569485930, 7.44886516548956788042446833261, 8.239930074767535175782349084199, 9.364685016844974211663849403086, 10.07342845999047413134696410902, 10.82423117934566306212380852075