| L(s) = 1 | + (−1.22 + 0.707i)2-s + (−1.94 − 2.28i)3-s + (0.999 − 1.73i)4-s + (7.34 − 4.24i)5-s + (4 + 1.41i)6-s + (−3.5 − 6.06i)7-s + 2.82i·8-s + (−1.39 + 8.89i)9-s + (−6 + 10.3i)10-s + (−5.89 + 1.09i)12-s − 13-s + (8.57 + 4.94i)14-s + (−24 − 8.48i)15-s + (−2.00 − 3.46i)16-s + (7.34 + 4.24i)17-s + (−4.57 − 11.8i)18-s + ⋯ |
| L(s) = 1 | + (−0.612 + 0.353i)2-s + (−0.649 − 0.760i)3-s + (0.249 − 0.433i)4-s + (1.46 − 0.848i)5-s + (0.666 + 0.235i)6-s + (−0.5 − 0.866i)7-s + 0.353i·8-s + (−0.155 + 0.987i)9-s + (−0.600 + 1.03i)10-s + (−0.491 + 0.0913i)12-s − 0.0769·13-s + (0.612 + 0.353i)14-s + (−1.60 − 0.565i)15-s + (−0.125 − 0.216i)16-s + (0.432 + 0.249i)17-s + (−0.254 − 0.659i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.600 + 0.799i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.600 + 0.799i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.711428 - 0.355498i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.711428 - 0.355498i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.22 - 0.707i)T \) |
| 3 | \( 1 + (1.94 + 2.28i)T \) |
| 7 | \( 1 + (3.5 + 6.06i)T \) |
| good | 5 | \( 1 + (-7.34 + 4.24i)T + (12.5 - 21.6i)T^{2} \) |
| 11 | \( 1 + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + T + 169T^{2} \) |
| 17 | \( 1 + (-7.34 - 4.24i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (-15.5 - 26.8i)T + (-180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (7.34 - 4.24i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 - 16.9iT - 841T^{2} \) |
| 31 | \( 1 + (-3.5 + 6.06i)T + (-480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + (-0.5 - 0.866i)T + (-684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 - 33.9iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 31T + 1.84e3T^{2} \) |
| 47 | \( 1 + (-36.7 + 21.2i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + (22.0 + 12.7i)T + (1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (-7.34 - 4.24i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (25 + 43.3i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (32.5 - 56.2i)T + (-2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 + 59.3iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (-48.5 + 84.0i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-51.5 - 89.2i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 - 42.4iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (-102. + 59.3i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + 166T + 9.40e3T^{2} \) |
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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
−16.39190213177610978220397372416, −14.16098159507275071260652467649, −13.34107312477895773002517520789, −12.24304222520835132215624480451, −10.46466730889875715909524166615, −9.592732951501568094383254296005, −7.928753997413634525307716955306, −6.45323873786755426348797049002, −5.40125617734530689683870714803, −1.39112087438542918662457635971,
2.80747555164711503271266000506, 5.52113227596535388078053963676, 6.70008383196147697966709853287, 9.190565567890237217673933638762, 9.834972473169451221305491644455, 10.89010979600188991984481954923, 12.11034276014547241106603649481, 13.62937298932252827348152191408, 15.04192788755563942695861491182, 16.12051704552349707216975597672
