L(s) = 1 | + (−1.28 + 1.16i)3-s + (−1.03 − 0.871i)5-s + (−4.32 − 1.57i)7-s + (0.296 − 2.98i)9-s + (0.700 − 0.587i)11-s + (0.939 − 5.32i)13-s + (2.34 − 0.0886i)15-s + (−3.81 + 6.60i)17-s + (−0.825 − 1.42i)19-s + (7.38 − 3.00i)21-s + (−5.39 + 1.96i)23-s + (−0.549 − 3.11i)25-s + (3.09 + 4.17i)27-s + (−0.698 − 3.96i)29-s + (−2.21 + 0.806i)31-s + ⋯ |
L(s) = 1 | + (−0.741 + 0.671i)3-s + (−0.464 − 0.389i)5-s + (−1.63 − 0.595i)7-s + (0.0987 − 0.995i)9-s + (0.211 − 0.177i)11-s + (0.260 − 1.47i)13-s + (0.605 − 0.0229i)15-s + (−0.925 + 1.60i)17-s + (−0.189 − 0.328i)19-s + (1.61 − 0.656i)21-s + (−1.12 + 0.409i)23-s + (−0.109 − 0.622i)25-s + (0.594 + 0.803i)27-s + (−0.129 − 0.735i)29-s + (−0.398 + 0.144i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.645 + 0.764i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.645 + 0.764i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.119244 - 0.256721i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.119244 - 0.256721i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.28 - 1.16i)T \) |
good | 5 | \( 1 + (1.03 + 0.871i)T + (0.868 + 4.92i)T^{2} \) |
| 7 | \( 1 + (4.32 + 1.57i)T + (5.36 + 4.49i)T^{2} \) |
| 11 | \( 1 + (-0.700 + 0.587i)T + (1.91 - 10.8i)T^{2} \) |
| 13 | \( 1 + (-0.939 + 5.32i)T + (-12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (3.81 - 6.60i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.825 + 1.42i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (5.39 - 1.96i)T + (17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (0.698 + 3.96i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (2.21 - 0.806i)T + (23.7 - 19.9i)T^{2} \) |
| 37 | \( 1 + (2.81 - 4.87i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (0.570 - 3.23i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (-3.36 + 2.82i)T + (7.46 - 42.3i)T^{2} \) |
| 47 | \( 1 + (-4.81 - 1.75i)T + (36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 - 2.84T + 53T^{2} \) |
| 59 | \( 1 + (0.651 + 0.547i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (3.48 + 1.26i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (-1.36 + 7.74i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (-6.18 + 10.7i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (2.00 + 3.46i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (2.30 + 13.0i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (0.992 + 5.62i)T + (-77.9 + 28.3i)T^{2} \) |
| 89 | \( 1 + (-4.63 - 8.02i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (8.62 - 7.23i)T + (16.8 - 95.5i)T^{2} \) |
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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
−12.08427442860053096728963565121, −10.71655988962895389886133029341, −10.26387244016045713958538187377, −9.210658276211916960116595113107, −8.029383732653524404859093254021, −6.51182835053080745690402128125, −5.86804585108886744761165196230, −4.24457945587408501828001520038, −3.44198080112277479901129951884, −0.24707724750500436222643380035,
2.39478876169601077753950553720, 4.03157036241662107524057473510, 5.65942939837331903071482041831, 6.74609715508008707399587679586, 7.13956191028816609322813391530, 8.858210353584304078843398809724, 9.704083350636149452530414482831, 11.00289420490632079436050851876, 11.81990051443442003854362892341, 12.47831385343152943740788189233