| L(s) = 1 | + (−0.637 + 0.112i)2-s + (−1.11 − 1.32i)3-s + (−3.36 + 1.22i)4-s + (−6.57 − 2.39i)5-s + (0.859 + 0.720i)6-s + (−1.12 + 1.95i)7-s + (4.25 − 2.45i)8-s + (−0.520 + 2.95i)9-s + (4.46 + 0.787i)10-s + (−5.72 − 9.92i)11-s + (5.37 + 3.10i)12-s + (−4.27 + 5.09i)13-s + (0.499 − 1.37i)14-s + (4.14 + 11.3i)15-s + (8.53 − 7.16i)16-s + (−2.16 − 12.2i)17-s + ⋯ |
| L(s) = 1 | + (−0.318 + 0.0562i)2-s + (−0.371 − 0.442i)3-s + (−0.841 + 0.306i)4-s + (−1.31 − 0.478i)5-s + (0.143 + 0.120i)6-s + (−0.161 + 0.279i)7-s + (0.531 − 0.306i)8-s + (−0.0578 + 0.328i)9-s + (0.446 + 0.0787i)10-s + (−0.520 − 0.902i)11-s + (0.447 + 0.258i)12-s + (−0.328 + 0.391i)13-s + (0.0356 − 0.0980i)14-s + (0.276 + 0.759i)15-s + (0.533 − 0.447i)16-s + (−0.127 − 0.720i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.976 + 0.214i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.976 + 0.214i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0163072 - 0.150083i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0163072 - 0.150083i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.11 + 1.32i)T \) |
| 19 | \( 1 + (-12.4 - 14.3i)T \) |
| good | 2 | \( 1 + (0.637 - 0.112i)T + (3.75 - 1.36i)T^{2} \) |
| 5 | \( 1 + (6.57 + 2.39i)T + (19.1 + 16.0i)T^{2} \) |
| 7 | \( 1 + (1.12 - 1.95i)T + (-24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (5.72 + 9.92i)T + (-60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (4.27 - 5.09i)T + (-29.3 - 166. i)T^{2} \) |
| 17 | \( 1 + (2.16 + 12.2i)T + (-271. + 98.8i)T^{2} \) |
| 23 | \( 1 + (20.3 - 7.39i)T + (405. - 340. i)T^{2} \) |
| 29 | \( 1 + (6.16 + 1.08i)T + (790. + 287. i)T^{2} \) |
| 31 | \( 1 + (44.9 + 25.9i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + 47.8iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (-4.93 - 5.87i)T + (-291. + 1.65e3i)T^{2} \) |
| 43 | \( 1 + (49.4 + 18.0i)T + (1.41e3 + 1.18e3i)T^{2} \) |
| 47 | \( 1 + (12.5 - 71.2i)T + (-2.07e3 - 755. i)T^{2} \) |
| 53 | \( 1 + (24.2 + 66.6i)T + (-2.15e3 + 1.80e3i)T^{2} \) |
| 59 | \( 1 + (-66.4 + 11.7i)T + (3.27e3 - 1.19e3i)T^{2} \) |
| 61 | \( 1 + (-108. + 39.3i)T + (2.85e3 - 2.39e3i)T^{2} \) |
| 67 | \( 1 + (64.2 + 11.3i)T + (4.21e3 + 1.53e3i)T^{2} \) |
| 71 | \( 1 + (35.7 - 98.2i)T + (-3.86e3 - 3.24e3i)T^{2} \) |
| 73 | \( 1 + (-81.9 + 68.7i)T + (925. - 5.24e3i)T^{2} \) |
| 79 | \( 1 + (-2.37 - 2.83i)T + (-1.08e3 + 6.14e3i)T^{2} \) |
| 83 | \( 1 + (-33.9 + 58.7i)T + (-3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + (64.2 - 76.5i)T + (-1.37e3 - 7.80e3i)T^{2} \) |
| 97 | \( 1 + (-71.7 + 12.6i)T + (8.84e3 - 3.21e3i)T^{2} \) |
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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
−14.33588056796864062254211413241, −13.14352499511356555632416220960, −12.19650258552213210461936905254, −11.26094235513362989494468687656, −9.512871919991449573034354999813, −8.255996923128524807308796770317, −7.48871406088154292552945606889, −5.38076769320746089987827101633, −3.82201614099165696230455230440, −0.17038282502028985804219824388,
3.81755941047252285258935327534, 5.07878812859442351318682879531, 7.17260948961516196665160169528, 8.368970057995563221032144215687, 9.888983817678814576683815723734, 10.70032946397570588140532956228, 11.95845304460742542214489147424, 13.19224961714257139436816513198, 14.72157509903701813010068335933, 15.37076056828041516453661340849
