| L(s) = 1 | + (1.22 − 0.707i)2-s + (0.518 − 0.898i)3-s + (0.999 − 1.73i)4-s + (−2.98 − 4.01i)5-s − 1.46i·6-s + (6.67 − 2.09i)7-s − 2.82i·8-s + (3.96 + 6.86i)9-s + (−6.49 − 2.80i)10-s + (−6.23 + 10.8i)11-s + (−1.03 − 1.79i)12-s − 0.748·13-s + (6.69 − 7.28i)14-s + (−5.15 + 0.601i)15-s + (−2.00 − 3.46i)16-s + (−5.77 + 10.0i)17-s + ⋯ |
| L(s) = 1 | + (0.612 − 0.353i)2-s + (0.172 − 0.299i)3-s + (0.249 − 0.433i)4-s + (−0.597 − 0.802i)5-s − 0.244i·6-s + (0.954 − 0.299i)7-s − 0.353i·8-s + (0.440 + 0.762i)9-s + (−0.649 − 0.280i)10-s + (−0.567 + 0.982i)11-s + (−0.0864 − 0.149i)12-s − 0.0575·13-s + (0.478 − 0.520i)14-s + (−0.343 + 0.0401i)15-s + (−0.125 − 0.216i)16-s + (−0.339 + 0.588i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.519 + 0.854i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.519 + 0.854i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.48851 - 0.836498i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.48851 - 0.836498i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.22 + 0.707i)T \) |
| 5 | \( 1 + (2.98 + 4.01i)T \) |
| 7 | \( 1 + (-6.67 + 2.09i)T \) |
| good | 3 | \( 1 + (-0.518 + 0.898i)T + (-4.5 - 7.79i)T^{2} \) |
| 11 | \( 1 + (6.23 - 10.8i)T + (-60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + 0.748T + 169T^{2} \) |
| 17 | \( 1 + (5.77 - 10.0i)T + (-144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (-13.0 + 7.56i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (31.4 - 18.1i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 - 50.5T + 841T^{2} \) |
| 31 | \( 1 + (40.4 + 23.3i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (-10.1 + 5.86i)T + (684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 - 12.7iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 26.2iT - 1.84e3T^{2} \) |
| 47 | \( 1 + (20.2 + 35.0i)T + (-1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + (22.8 + 13.1i)T + (1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (68.4 + 39.5i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-45.2 + 26.1i)T + (1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (30.0 + 17.3i)T + (2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 - 14.6T + 5.04e3T^{2} \) |
| 73 | \( 1 + (50.5 - 87.6i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-17.8 - 30.9i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + 80.8T + 6.88e3T^{2} \) |
| 89 | \( 1 + (117. - 67.6i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 - 91.4T + 9.40e3T^{2} \) |
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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
−14.06304958792675581737594448891, −13.12982386068258260948805980324, −12.22857849889279572413608766548, −11.18192512995799546878727654923, −9.950181412244409031511867313701, −8.205862763560643925057345116320, −7.35299543996687389821136219556, −5.18692742968335179151968349526, −4.25218017487944130811365025174, −1.82874745712578982519103496432,
3.06803081011460566430822903834, 4.51439728826382942146095052291, 6.13127792739619679615378939632, 7.52516477700073139457989659690, 8.596028780379700538959714071018, 10.35877732773428225212351395662, 11.47078123492753043191368245592, 12.34601669120327053662267000471, 14.00912016129974711144709516876, 14.52376428461943849543560999593
