L(s) = 1 | + (1.81 − 3.13i)5-s + (1.59 − 0.920i)7-s + (−10.0 + 5.80i)11-s + (−6.43 + 11.1i)13-s − 12.6·17-s − 25.6i·19-s + (25.9 + 14.9i)23-s + (5.93 + 10.2i)25-s + (−10.8 − 18.7i)29-s + (52.4 + 30.2i)31-s − 6.67i·35-s + 25.7·37-s + (33.3 − 57.7i)41-s + (14.2 − 8.22i)43-s + (66.1 − 38.1i)47-s + ⋯ |
L(s) = 1 | + (0.362 − 0.627i)5-s + (0.227 − 0.131i)7-s + (−0.914 + 0.528i)11-s + (−0.495 + 0.857i)13-s − 0.742·17-s − 1.35i·19-s + (1.12 + 0.650i)23-s + (0.237 + 0.411i)25-s + (−0.372 − 0.645i)29-s + (1.69 + 0.975i)31-s − 0.190i·35-s + 0.695·37-s + (0.813 − 1.40i)41-s + (0.331 − 0.191i)43-s + (1.40 − 0.812i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.864 + 0.503i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.864 + 0.503i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.964619409\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.964619409\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-1.81 + 3.13i)T + (-12.5 - 21.6i)T^{2} \) |
| 7 | \( 1 + (-1.59 + 0.920i)T + (24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (10.0 - 5.80i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (6.43 - 11.1i)T + (-84.5 - 146. i)T^{2} \) |
| 17 | \( 1 + 12.6T + 289T^{2} \) |
| 19 | \( 1 + 25.6iT - 361T^{2} \) |
| 23 | \( 1 + (-25.9 - 14.9i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (10.8 + 18.7i)T + (-420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (-52.4 - 30.2i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 - 25.7T + 1.36e3T^{2} \) |
| 41 | \( 1 + (-33.3 + 57.7i)T + (-840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (-14.2 + 8.22i)T + (924.5 - 1.60e3i)T^{2} \) |
| 47 | \( 1 + (-66.1 + 38.1i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 - 14.2T + 2.80e3T^{2} \) |
| 59 | \( 1 + (50.3 + 29.0i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-9.43 - 16.3i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-20.6 - 11.9i)T + (2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 + 46.4iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 49.3T + 5.32e3T^{2} \) |
| 79 | \( 1 + (-52.4 + 30.2i)T + (3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-86.2 + 49.8i)T + (3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + 154.T + 7.92e3T^{2} \) |
| 97 | \( 1 + (21.1 + 36.5i)T + (-4.70e3 + 8.14e3i)T^{2} \) |
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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
−9.183809234286829859120903548480, −8.404250319455202542236882434438, −7.32905960950875839461443406851, −6.89106192987949341295825131612, −5.67649303970706115441867388754, −4.83528032629709687021016731059, −4.40264473817614672931737178364, −2.85534269939927656665684892853, −2.00841783542793063220924733892, −0.69506244724675054245724537603,
0.850505606236439137957866764355, 2.47268499282903299403602960536, 2.90915787475936807755259834405, 4.26051421360793003246482822513, 5.20284358969222445400552341614, 6.00874547186640250443072176475, 6.70319997482219158576656944408, 7.85128006409679363644485052101, 8.164971762144350989467182222663, 9.276558115388382841433061073790