L(s) = 1 | + (−0.866 − 0.5i)2-s + (0.499 + 0.866i)4-s + (−0.995 − 1.72i)5-s − 0.999i·8-s + 1.99i·10-s + (−5.21 − 3.00i)11-s + (−3.43 + 1.98i)13-s + (−0.5 + 0.866i)16-s − 1.56·17-s − 4.80i·19-s + (0.995 − 1.72i)20-s + (3.00 + 5.21i)22-s + (5.02 − 2.90i)23-s + (0.519 − 0.899i)25-s + 3.96·26-s + ⋯ |
L(s) = 1 | + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s + (−0.445 − 0.770i)5-s − 0.353i·8-s + 0.629i·10-s + (−1.57 − 0.907i)11-s + (−0.951 + 0.549i)13-s + (−0.125 + 0.216i)16-s − 0.378·17-s − 1.10i·19-s + (0.222 − 0.385i)20-s + (0.641 + 1.11i)22-s + (1.04 − 0.604i)23-s + (0.103 − 0.179i)25-s + 0.777·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.301 - 0.953i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.301 - 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2155555163\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2155555163\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 + 0.5i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (0.995 + 1.72i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (5.21 + 3.00i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (3.43 - 1.98i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + 1.56T + 17T^{2} \) |
| 19 | \( 1 + 4.80iT - 19T^{2} \) |
| 23 | \( 1 + (-5.02 + 2.90i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (5.26 + 3.03i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (1.11 - 0.643i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 6.05T + 37T^{2} \) |
| 41 | \( 1 + (-2.98 - 5.16i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (4.53 - 7.85i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (2.39 - 4.14i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 8.39iT - 53T^{2} \) |
| 59 | \( 1 + (-6.38 - 11.0i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.61 - 0.933i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (3.79 + 6.57i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 5.48iT - 71T^{2} \) |
| 73 | \( 1 - 10.8iT - 73T^{2} \) |
| 79 | \( 1 + (-6.94 + 12.0i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-3.35 + 5.81i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 4.11T + 89T^{2} \) |
| 97 | \( 1 + (14.1 + 8.14i)T + (48.5 + 84.0i)T^{2} \) |
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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
−9.052570617671219578612254929178, −8.275193485411471297289714318312, −7.73795158347731484375033589995, −6.97325192346973120321913873102, −5.92769185270795798424901832385, −4.87298482330472769517564322554, −4.43507823648198196696606263964, −2.99003549518396211716251148771, −2.43490925177918148445552266838, −0.877863407193608830098095401661,
0.10810404202795911139170531887, 1.94367203060954229576523345955, 2.78813482944261925079505326486, 3.78392078510508212703211334295, 5.16040085196230191206002232081, 5.42808924957161561206999581175, 6.79726952428002827685634921924, 7.30421433632077311125664517863, 7.77940499562124547070443487295, 8.526070347417691827427326993773