L(s) = 1 | + (−1.5 − 0.866i)3-s + (1.5 − 2.59i)5-s + (2 − 1.73i)7-s + (1.5 + 2.59i)9-s + (−4.5 + 2.59i)11-s + (−4.5 + 2.59i)15-s + (1.5 + 2.59i)17-s + (1.5 + 0.866i)19-s + (−4.5 + 0.866i)21-s + (4.5 + 2.59i)23-s + (−2 − 3.46i)25-s − 5.19i·27-s + (−1.5 + 0.866i)31-s + 9·33-s + (−1.5 − 7.79i)35-s + ⋯ |
L(s) = 1 | + (−0.866 − 0.499i)3-s + (0.670 − 1.16i)5-s + (0.755 − 0.654i)7-s + (0.5 + 0.866i)9-s + (−1.35 + 0.783i)11-s + (−1.16 + 0.670i)15-s + (0.363 + 0.630i)17-s + (0.344 + 0.198i)19-s + (−0.981 + 0.188i)21-s + (0.938 + 0.541i)23-s + (−0.400 − 0.692i)25-s − 0.999i·27-s + (−0.269 + 0.155i)31-s + 1.56·33-s + (−0.253 − 1.31i)35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.553 + 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.553 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.748335 - 0.400992i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.748335 - 0.400992i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.5 + 0.866i)T \) |
| 7 | \( 1 + (-2 + 1.73i)T \) |
good | 5 | \( 1 + (-1.5 + 2.59i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (4.5 - 2.59i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + (-1.5 - 2.59i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.5 - 0.866i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-4.5 - 2.59i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + (1.5 - 0.866i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (3.5 - 6.06i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 + (-1.5 + 2.59i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-4.5 + 2.59i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (1.5 + 2.59i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (10.5 + 6.06i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (2.5 + 4.33i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 10.3iT - 71T^{2} \) |
| 73 | \( 1 + (10.5 - 6.06i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.5 + 0.866i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 12T + 83T^{2} \) |
| 89 | \( 1 + (4.5 - 7.79i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 6.92iT - 97T^{2} \) |
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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
−13.69169966142622659909948882410, −13.01658424383537857920447951262, −12.16075189994791473413079363820, −10.83467187824897371894014100415, −9.918395853026315812525942744810, −8.277117826161875237870392633277, −7.22703844650902517333582839116, −5.48565986050273276885501079767, −4.79948549506521339838917428403, −1.54619984643980010301992483740,
2.85316685493529774338497922753, 5.09949887049444236650262658522, 5.97543486726493630289169416536, 7.39495330944302374714503174829, 9.075181555363793739476404870973, 10.47296983286332629393831550726, 10.92505200911177285641596375675, 12.05639591848514170888361517975, 13.47331276317274255234512181947, 14.59174071179486161604584992940