L(s) = 1 | + (−1.49 + 2.40i)2-s + (4.65 − 8.06i)3-s + (−3.54 − 7.16i)4-s + (5.11 − 2.95i)5-s + (12.4 + 23.2i)6-s + (1.92 + 18.4i)7-s + (22.5 + 2.16i)8-s + (−29.8 − 51.6i)9-s + (−0.533 + 16.6i)10-s + (0.267 + 0.154i)11-s + (−74.3 − 4.75i)12-s + 43.7i·13-s + (−47.1 − 22.8i)14-s − 54.9i·15-s + (−38.8 + 50.8i)16-s + (−27.0 − 15.6i)17-s + ⋯ |
L(s) = 1 | + (−0.527 + 0.849i)2-s + (0.895 − 1.55i)3-s + (−0.443 − 0.896i)4-s + (0.457 − 0.264i)5-s + (0.845 + 1.57i)6-s + (0.104 + 0.994i)7-s + (0.995 + 0.0957i)8-s + (−1.10 − 1.91i)9-s + (−0.0168 + 0.528i)10-s + (0.00733 + 0.00423i)11-s + (−1.78 − 0.114i)12-s + 0.932i·13-s + (−0.899 − 0.436i)14-s − 0.946i·15-s + (−0.606 + 0.795i)16-s + (−0.386 − 0.223i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.937 + 0.348i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.937 + 0.348i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.16747 - 0.210295i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.16747 - 0.210295i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.49 - 2.40i)T \) |
| 7 | \( 1 + (-1.92 - 18.4i)T \) |
good | 3 | \( 1 + (-4.65 + 8.06i)T + (-13.5 - 23.3i)T^{2} \) |
| 5 | \( 1 + (-5.11 + 2.95i)T + (62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (-0.267 - 0.154i)T + (665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 - 43.7iT - 2.19e3T^{2} \) |
| 17 | \( 1 + (27.0 + 15.6i)T + (2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-39.7 - 68.8i)T + (-3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-16.5 + 9.52i)T + (6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + 40.1T + 2.43e4T^{2} \) |
| 31 | \( 1 + (-42.8 + 74.2i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-72.4 - 125. i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + 254. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + 366. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + (5.12 + 8.87i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-315. + 546. i)T + (-7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + (257. - 446. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (669. - 386. i)T + (1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (450. + 259. i)T + (1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + 261. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (-588. - 339. i)T + (1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (299. - 172. i)T + (2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + 805.T + 5.71e5T^{2} \) |
| 89 | \( 1 + (40.7 - 23.5i)T + (3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + 763. iT - 9.12e5T^{2} \) |
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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
−16.91553226582937517823234732702, −15.27421627409124043202434934339, −14.16899957519633373410932761819, −13.31963948701683024973550712031, −11.92137027181062949954406727758, −9.365305059021607187683573784276, −8.505329504612691334170171960371, −7.19826754142350639093327660683, −5.87496551643793317580591025453, −1.84803286112979944436215733835,
3.03172723846557677740783996652, 4.51104217868518588117348923760, 7.914949599242828351069300705835, 9.310817461839901129338068122541, 10.25025952938918426094034559446, 11.02615011502179764057834159661, 13.28502955708446515945157017154, 14.26307259970466111769431101296, 15.67130281095808996335305879607, 16.86030542831923439594164351650