L(s) = 1 | + (0.835 − 1.14i)2-s + (−0.604 − 1.90i)4-s + (0.823 − 0.341i)5-s + (0.760 − 0.760i)7-s + (−2.68 − 0.903i)8-s + (0.298 − 1.22i)10-s + (−1.75 − 4.24i)11-s + (2.78 + 1.15i)13-s + (−0.232 − 1.50i)14-s + (−3.26 + 2.30i)16-s − 0.00932i·17-s + (5.33 + 2.21i)19-s + (−1.14 − 1.36i)20-s + (−6.31 − 1.54i)22-s + (−2.06 − 2.06i)23-s + ⋯ |
L(s) = 1 | + (0.590 − 0.806i)2-s + (−0.302 − 0.953i)4-s + (0.368 − 0.152i)5-s + (0.287 − 0.287i)7-s + (−0.947 − 0.319i)8-s + (0.0944 − 0.387i)10-s + (−0.530 − 1.28i)11-s + (0.773 + 0.320i)13-s + (−0.0621 − 0.401i)14-s + (−0.817 + 0.575i)16-s − 0.00226i·17-s + (1.22 + 0.507i)19-s + (−0.256 − 0.305i)20-s + (−1.34 − 0.328i)22-s + (−0.430 − 0.430i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.212 + 0.977i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.212 + 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.09054 - 1.35378i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.09054 - 1.35378i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.835 + 1.14i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-0.823 + 0.341i)T + (3.53 - 3.53i)T^{2} \) |
| 7 | \( 1 + (-0.760 + 0.760i)T - 7iT^{2} \) |
| 11 | \( 1 + (1.75 + 4.24i)T + (-7.77 + 7.77i)T^{2} \) |
| 13 | \( 1 + (-2.78 - 1.15i)T + (9.19 + 9.19i)T^{2} \) |
| 17 | \( 1 + 0.00932iT - 17T^{2} \) |
| 19 | \( 1 + (-5.33 - 2.21i)T + (13.4 + 13.4i)T^{2} \) |
| 23 | \( 1 + (2.06 + 2.06i)T + 23iT^{2} \) |
| 29 | \( 1 + (1.24 - 2.99i)T + (-20.5 - 20.5i)T^{2} \) |
| 31 | \( 1 + 1.86T + 31T^{2} \) |
| 37 | \( 1 + (-4.94 + 2.04i)T + (26.1 - 26.1i)T^{2} \) |
| 41 | \( 1 + (-7.21 - 7.21i)T + 41iT^{2} \) |
| 43 | \( 1 + (-2.68 - 6.49i)T + (-30.4 + 30.4i)T^{2} \) |
| 47 | \( 1 + 8.24iT - 47T^{2} \) |
| 53 | \( 1 + (-4.21 - 10.1i)T + (-37.4 + 37.4i)T^{2} \) |
| 59 | \( 1 + (4.98 - 2.06i)T + (41.7 - 41.7i)T^{2} \) |
| 61 | \( 1 + (-4.20 + 10.1i)T + (-43.1 - 43.1i)T^{2} \) |
| 67 | \( 1 + (1.34 - 3.23i)T + (-47.3 - 47.3i)T^{2} \) |
| 71 | \( 1 + (5.86 - 5.86i)T - 71iT^{2} \) |
| 73 | \( 1 + (1.26 + 1.26i)T + 73iT^{2} \) |
| 79 | \( 1 + 13.6iT - 79T^{2} \) |
| 83 | \( 1 + (-6.49 - 2.69i)T + (58.6 + 58.6i)T^{2} \) |
| 89 | \( 1 + (10.7 - 10.7i)T - 89iT^{2} \) |
| 97 | \( 1 + 16.8T + 97T^{2} \) |
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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
−11.36020009768545097927248940341, −10.92734896318105580797813860942, −9.823617418945391605306979616362, −8.936236681516433837249703469221, −7.76553065118794721556075054155, −6.12388487328814131368087672561, −5.46842489768645327985566055719, −4.10531669680177765968432008414, −2.96660881353489350727768300441, −1.27658826632230467080868892598,
2.43177265465671022184911926584, 3.96640127859181809814277021058, 5.17818711480300947488088369773, 5.98699865279194590285188504528, 7.22814500966120240818566551379, 7.939301476040601954640064482093, 9.130700366875050908896126200358, 10.05159602133178222845938245316, 11.38876631128095975115037620498, 12.25054139441790186133069978056