| L(s) = 1 | + (−2.47 − 0.526i)2-s + (−1.29 + 1.15i)3-s + (4.03 + 1.79i)4-s + (1.67 − 1.48i)5-s + (3.81 − 2.16i)6-s + (−1.03 + 1.79i)7-s + (−4.94 − 3.59i)8-s + (0.350 − 2.97i)9-s + (−4.92 + 2.80i)10-s + (−2.39 − 0.509i)11-s + (−7.28 + 2.31i)12-s + (1.74 − 0.371i)13-s + (3.50 − 3.89i)14-s + (−0.452 + 3.84i)15-s + (4.44 + 4.94i)16-s + (3.67 + 2.66i)17-s + ⋯ |
| L(s) = 1 | + (−1.75 − 0.372i)2-s + (−0.747 + 0.664i)3-s + (2.01 + 0.897i)4-s + (0.747 − 0.664i)5-s + (1.55 − 0.885i)6-s + (−0.390 + 0.676i)7-s + (−1.74 − 1.26i)8-s + (0.116 − 0.993i)9-s + (−1.55 + 0.885i)10-s + (−0.722 − 0.153i)11-s + (−2.10 + 0.668i)12-s + (0.484 − 0.102i)13-s + (0.936 − 1.04i)14-s + (−0.116 + 0.993i)15-s + (1.11 + 1.23i)16-s + (0.891 + 0.647i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 - 0.472i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.881 - 0.472i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.456025 + 0.114427i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.456025 + 0.114427i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.29 - 1.15i)T \) |
| 5 | \( 1 + (-1.67 + 1.48i)T \) |
| good | 2 | \( 1 + (2.47 + 0.526i)T + (1.82 + 0.813i)T^{2} \) |
| 7 | \( 1 + (1.03 - 1.79i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (2.39 + 0.509i)T + (10.0 + 4.47i)T^{2} \) |
| 13 | \( 1 + (-1.74 + 0.371i)T + (11.8 - 5.28i)T^{2} \) |
| 17 | \( 1 + (-3.67 - 2.66i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (-5.85 - 4.25i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (-1.58 + 1.75i)T + (-2.40 - 22.8i)T^{2} \) |
| 29 | \( 1 + (-0.855 - 8.13i)T + (-28.3 + 6.02i)T^{2} \) |
| 31 | \( 1 + (0.709 - 6.74i)T + (-30.3 - 6.44i)T^{2} \) |
| 37 | \( 1 + (-1.53 + 4.73i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (-4.68 + 0.995i)T + (37.4 - 16.6i)T^{2} \) |
| 43 | \( 1 + (-4.14 + 7.17i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-1.05 - 10.0i)T + (-45.9 + 9.77i)T^{2} \) |
| 53 | \( 1 + (2.52 - 1.83i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (10.2 - 2.17i)T + (53.8 - 23.9i)T^{2} \) |
| 61 | \( 1 + (-2.57 - 0.548i)T + (55.7 + 24.8i)T^{2} \) |
| 67 | \( 1 + (-0.170 + 1.62i)T + (-65.5 - 13.9i)T^{2} \) |
| 71 | \( 1 + (-8.24 + 5.98i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (0.608 + 1.87i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (1.55 + 14.8i)T + (-77.2 + 16.4i)T^{2} \) |
| 83 | \( 1 + (-3.70 + 1.65i)T + (55.5 - 61.6i)T^{2} \) |
| 89 | \( 1 + (-0.105 - 0.326i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (-1.15 - 10.9i)T + (-94.8 + 20.1i)T^{2} \) |
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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
−12.25111556557661049973649842053, −10.79499701896827361778627809002, −10.37264379734685469245952919670, −9.362989314057613446765116276665, −8.885997463423283558030770654234, −7.67510081974011972612317841798, −6.17140885690663740883443589922, −5.34249557875273612518377990678, −3.13801780513246908978135179245, −1.24924017804354877153219695777,
0.905369034657321267552485791033, 2.57290968824194517097890269377, 5.48882783992019169058045351198, 6.46974501799522182227595865812, 7.30442377893333359271848317523, 7.894604997789627436702431697799, 9.572998604859615837044511625731, 9.979180100663370330183351105602, 11.02571109590368636374857653269, 11.57480147460216937370261901147
