L(s) = 1 | + (−2.89 − 0.783i)3-s + (2.35 − 4.08i)7-s + (7.77 + 4.53i)9-s + (16.6 + 9.63i)11-s + (−4.68 − 8.11i)13-s + 16.2i·17-s − 19.0·19-s + (−10.0 + 9.97i)21-s + (−14.3 + 8.30i)23-s + (−18.9 − 19.2i)27-s + (−3.87 − 2.23i)29-s + (−7.19 − 12.4i)31-s + (−40.7 − 40.9i)33-s + 55.6·37-s + (7.21 + 27.1i)39-s + ⋯ |
L(s) = 1 | + (−0.965 − 0.261i)3-s + (0.336 − 0.583i)7-s + (0.863 + 0.504i)9-s + (1.51 + 0.875i)11-s + (−0.360 − 0.624i)13-s + 0.955i·17-s − 1.00·19-s + (−0.477 + 0.474i)21-s + (−0.625 + 0.361i)23-s + (−0.702 − 0.712i)27-s + (−0.133 − 0.0771i)29-s + (−0.232 − 0.401i)31-s + (−1.23 − 1.24i)33-s + 1.50·37-s + (0.184 + 0.696i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.941 - 0.337i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.941 - 0.337i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.378549224\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.378549224\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (2.89 + 0.783i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (-2.35 + 4.08i)T + (-24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (-16.6 - 9.63i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (4.68 + 8.11i)T + (-84.5 + 146. i)T^{2} \) |
| 17 | \( 1 - 16.2iT - 289T^{2} \) |
| 19 | \( 1 + 19.0T + 361T^{2} \) |
| 23 | \( 1 + (14.3 - 8.30i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (3.87 + 2.23i)T + (420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (7.19 + 12.4i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 - 55.6T + 1.36e3T^{2} \) |
| 41 | \( 1 + (-26.7 + 15.4i)T + (840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (21.5 - 37.3i)T + (-924.5 - 1.60e3i)T^{2} \) |
| 47 | \( 1 + (-35.9 - 20.7i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 - 30.7iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (-74.4 + 42.9i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (51.7 - 89.6i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-61.7 - 107. i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 + 111. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 90.5T + 5.32e3T^{2} \) |
| 79 | \( 1 + (-41.6 + 72.1i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (31.0 + 17.9i)T + (3.44e3 + 5.96e3i)T^{2} \) |
| 89 | \( 1 - 84.9iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (52.6 - 91.1i)T + (-4.70e3 - 8.14e3i)T^{2} \) |
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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
−10.08997758928284535195094555037, −9.312947391232901374533007780056, −8.043635227613505919938705335938, −7.33563467814036001091319456433, −6.44641580261524826425235270758, −5.78451624162627955183510678230, −4.45452132190577529106016178354, −4.02829047883255958719980014686, −2.04854420156833279352142334515, −0.978444118735366540009231876771,
0.67073357700267167684823409713, 2.09622234899349444037734120589, 3.73467176520999964220148260920, 4.55326031730789099237215474427, 5.55493066286171685835751998021, 6.38136100748556432623289894905, 6.99756154589835198895548065603, 8.343196295024637501813784170860, 9.161605884648978815632923854006, 9.794850604875108800162761198788