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
\(L(1)\) |
\(\approx\) |
\(1.19770 - 0.838640i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.19770 - 0.838640i\) |
\(L(\frac{3}{2})\) |
|
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
−10.08247948867307977265447116529, −8.948167377182248607137867456958, −8.491437906248438540966976744119, −7.41220702760599171596600794502, −6.63342347962550442990517033452, −5.59286113353207558837588242471, −4.45749947476653311833194925548, −3.81183399865571747924163200392, −2.27587100294022023301759205155, −0.76582098191944254966077180231,
1.52362582146121288986838691382, 2.91513469288904632213006676413, 3.87471714850745977816084005231, 5.17606758144977147600303497265, 5.83534420884868904694194130286, 7.10299544640318229107168666735, 7.63320597480559284817936142171, 8.747875263487756937319423255493, 9.399216222603776714250181051300, 10.37616385167495920860524985394