| L(s) = 1 | + (−0.366 + 1.36i)2-s + (−0.636 − 2.37i)3-s + (−1.73 − i)4-s + (4.96 + 0.598i)5-s + 3.47·6-s + (6.99 + 0.219i)7-s + (2 − 1.99i)8-s + (2.55 − 1.47i)9-s + (−2.63 + 6.56i)10-s + (3.73 − 6.47i)11-s + (−1.27 + 4.74i)12-s + (−10.9 + 10.9i)13-s + (−2.86 + 9.47i)14-s + (−1.73 − 12.1i)15-s + (1.99 + 3.46i)16-s + (−20.4 + 5.47i)17-s + ⋯ |
| L(s) = 1 | + (−0.183 + 0.683i)2-s + (−0.212 − 0.791i)3-s + (−0.433 − 0.250i)4-s + (0.992 + 0.119i)5-s + 0.579·6-s + (0.999 + 0.0313i)7-s + (0.250 − 0.249i)8-s + (0.284 − 0.164i)9-s + (−0.263 + 0.656i)10-s + (0.339 − 0.588i)11-s + (−0.106 + 0.395i)12-s + (−0.842 + 0.842i)13-s + (−0.204 + 0.676i)14-s + (−0.115 − 0.811i)15-s + (0.124 + 0.216i)16-s + (−1.20 + 0.321i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0770i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.997 - 0.0770i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.24426 + 0.0480322i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.24426 + 0.0480322i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.366 - 1.36i)T \) |
| 5 | \( 1 + (-4.96 - 0.598i)T \) |
| 7 | \( 1 + (-6.99 - 0.219i)T \) |
| good | 3 | \( 1 + (0.636 + 2.37i)T + (-7.79 + 4.5i)T^{2} \) |
| 11 | \( 1 + (-3.73 + 6.47i)T + (-60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (10.9 - 10.9i)T - 169iT^{2} \) |
| 17 | \( 1 + (20.4 - 5.47i)T + (250. - 144.5i)T^{2} \) |
| 19 | \( 1 + (-12.4 + 7.21i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (20.0 + 5.37i)T + (458. + 264.5i)T^{2} \) |
| 29 | \( 1 - 15.8iT - 841T^{2} \) |
| 31 | \( 1 + (8 - 13.8i)T + (-480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + (17.1 - 64.1i)T + (-1.18e3 - 684.5i)T^{2} \) |
| 41 | \( 1 + 27.9T + 1.68e3T^{2} \) |
| 43 | \( 1 + (-39.6 + 39.6i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (18.6 - 69.4i)T + (-1.91e3 - 1.10e3i)T^{2} \) |
| 53 | \( 1 + (10.9 + 40.9i)T + (-2.43e3 + 1.40e3i)T^{2} \) |
| 59 | \( 1 + (55.8 + 32.2i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (29.5 + 51.1i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-53.5 + 14.3i)T + (3.88e3 - 2.24e3i)T^{2} \) |
| 71 | \( 1 + 36.3T + 5.04e3T^{2} \) |
| 73 | \( 1 + (-17.1 - 63.8i)T + (-4.61e3 + 2.66e3i)T^{2} \) |
| 79 | \( 1 + (-35.6 + 20.6i)T + (3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (9.12 - 9.12i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 + (42.1 - 24.3i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (63.7 + 63.7i)T + 9.40e3iT^{2} \) |
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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
−14.23348619267203529656316594567, −13.72107805600083302100546001788, −12.42713066026873927546774841568, −11.17706095552423433688680668143, −9.718980437988453329421945142102, −8.568813757014022683560833467011, −7.14046707587119366902327529667, −6.26892697657306487656914490794, −4.79932166277065310412927456234, −1.72104592906587766781228842942,
2.04742508486996127144459820720, 4.37327180362109069719018696304, 5.41807094129309623284154270673, 7.55698888888795034959258547836, 9.168835925990677561861756122743, 10.02703586460510369447354420912, 10.85576958668356754369023114462, 12.09605353999749198079920329151, 13.30749151670037891899117624814, 14.37558238200573244656256250578
