L(s) = 1 | + (0.866 + 1.5i)2-s + (0.724 + 1.57i)3-s + (−0.5 + 0.866i)4-s + (0.358 − 2.20i)5-s + (−1.73 + 2.44i)6-s + 1.73·8-s + (−1.94 + 2.28i)9-s + (3.62 − 1.37i)10-s + (2.44 + 1.41i)11-s + (−1.72 − 0.158i)12-s + 4·13-s + (3.73 − 1.03i)15-s + (2.49 + 4.33i)16-s + (−2.44 − 1.41i)17-s + (−5.10 − 0.949i)18-s + ⋯ |
L(s) = 1 | + (0.612 + 1.06i)2-s + (0.418 + 0.908i)3-s + (−0.250 + 0.433i)4-s + (0.160 − 0.987i)5-s + (−0.707 + 0.999i)6-s + 0.612·8-s + (−0.649 + 0.760i)9-s + (1.14 − 0.434i)10-s + (0.738 + 0.426i)11-s + (−0.497 − 0.0458i)12-s + 1.10·13-s + (0.963 − 0.267i)15-s + (0.624 + 1.08i)16-s + (−0.594 − 0.342i)17-s + (−1.20 − 0.223i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.205 - 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.205 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.71793 + 2.11666i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.71793 + 2.11666i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.724 - 1.57i)T \) |
| 5 | \( 1 + (-0.358 + 2.20i)T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (-0.866 - 1.5i)T + (-1 + 1.73i)T^{2} \) |
| 11 | \( 1 + (-2.44 - 1.41i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 4T + 13T^{2} \) |
| 17 | \( 1 + (2.44 + 1.41i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.73 - 3i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 5.65iT - 29T^{2} \) |
| 31 | \( 1 + (8.48 + 4.89i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 3.46T + 41T^{2} \) |
| 43 | \( 1 + 4.89iT - 43T^{2} \) |
| 47 | \( 1 + (-2.44 + 1.41i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.46 + 6i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (8.48 - 4.89i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (4.24 + 2.44i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 2.82iT - 71T^{2} \) |
| 73 | \( 1 + (-4 + 6.92i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (4 + 6.92i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 2.82iT - 83T^{2} \) |
| 89 | \( 1 + (-5.19 - 9i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 8T + 97T^{2} \) |
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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
−10.61209635589324081199130768962, −9.381415738071790960731790451653, −8.974083533910311817357427827965, −8.038745875652973587577703574245, −7.07332549810623077061408416880, −5.93430631177754586338892968637, −5.25630454875332259232597679959, −4.37757962405988752325907230917, −3.67518180464445118210757641545, −1.71761675171600041691113280698,
1.37530460747743157402968343138, 2.43847119916631803215484171248, 3.35035653529814721748392111824, 4.09475637191309815281492426865, 5.82575900359298957649952101241, 6.59200307922928769753017485506, 7.43188377942543029943182344630, 8.446762471742849798705196225556, 9.349280930957885025951491865586, 10.56463840414735325221855699901