L(s) = 1 | + (−0.5 + 0.866i)5-s + (−0.866 − 1.5i)7-s + (−0.866 − 1.5i)11-s + (1.5 − 2.59i)13-s − 4·17-s − 6.92·19-s + (4.33 − 7.5i)23-s + (2 + 3.46i)25-s + (0.5 + 0.866i)29-s + (2.59 − 4.5i)31-s + 1.73·35-s − 8·37-s + (2.5 − 4.33i)41-s + (−4.33 − 7.5i)43-s + (−6.06 − 10.5i)47-s + ⋯ |
L(s) = 1 | + (−0.223 + 0.387i)5-s + (−0.327 − 0.566i)7-s + (−0.261 − 0.452i)11-s + (0.416 − 0.720i)13-s − 0.970·17-s − 1.58·19-s + (0.902 − 1.56i)23-s + (0.400 + 0.692i)25-s + (0.0928 + 0.160i)29-s + (0.466 − 0.808i)31-s + 0.292·35-s − 1.31·37-s + (0.390 − 0.676i)41-s + (−0.660 − 1.14i)43-s + (−0.884 − 1.53i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.342 + 0.939i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.342 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.498156 - 0.711441i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.498156 - 0.711441i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (0.5 - 0.866i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (0.866 + 1.5i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (0.866 + 1.5i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.5 + 2.59i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + 4T + 17T^{2} \) |
| 19 | \( 1 + 6.92T + 19T^{2} \) |
| 23 | \( 1 + (-4.33 + 7.5i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.5 - 0.866i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-2.59 + 4.5i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 8T + 37T^{2} \) |
| 41 | \( 1 + (-2.5 + 4.33i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (4.33 + 7.5i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (6.06 + 10.5i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 8T + 53T^{2} \) |
| 59 | \( 1 + (0.866 - 1.5i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-3.5 - 6.06i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (4.33 - 7.5i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 3.46T + 71T^{2} \) |
| 73 | \( 1 + 12T + 73T^{2} \) |
| 79 | \( 1 + (2.59 + 4.5i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-4.33 - 7.5i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 4T + 89T^{2} \) |
| 97 | \( 1 + (-1.5 - 2.59i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.33575090294345762202395190773, −8.698110722396854855391143073714, −8.533982072457062344870891010477, −7.09853843779683272014800045371, −6.67241522092680451373283018568, −5.55114412487907465686362515561, −4.38955312164940645350925989657, −3.46360293172836975393522441281, −2.33765199261788663464985792543, −0.40763888665718253809581221461,
1.69301199559763503341497573112, 2.93912138126974346374581079361, 4.25708785557643389066713286770, 4.97779361290707158142621033078, 6.23629546583189243081733952079, 6.83849266015009819921808547985, 8.037052249607550983392157739790, 8.832101441165203706592533616484, 9.374827109736983308141427238316, 10.47000541267227582098609690178