L(s) = 1 | − i·2-s − 4-s + 2i·7-s + i·8-s + 4i·13-s + 2·14-s + 16-s + 6i·17-s + 7·19-s + 4·26-s − 2i·28-s − 6·29-s − 10·31-s − i·32-s + 6·34-s + ⋯ |
L(s) = 1 | − 0.707i·2-s − 0.5·4-s + 0.755i·7-s + 0.353i·8-s + 1.10i·13-s + 0.534·14-s + 0.250·16-s + 1.45i·17-s + 1.60·19-s + 0.784·26-s − 0.377i·28-s − 1.11·29-s − 1.79·31-s − 0.176i·32-s + 1.02·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7526324455\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7526324455\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 2iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 4iT - 13T^{2} \) |
| 17 | \( 1 - 6iT - 17T^{2} \) |
| 19 | \( 1 - 7T + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + 10T + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 + 9T + 41T^{2} \) |
| 43 | \( 1 - iT - 43T^{2} \) |
| 47 | \( 1 + 6iT - 47T^{2} \) |
| 53 | \( 1 + 12iT - 53T^{2} \) |
| 59 | \( 1 + 9T + 59T^{2} \) |
| 61 | \( 1 + 4T + 61T^{2} \) |
| 67 | \( 1 + 13iT - 67T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 - iT - 73T^{2} \) |
| 79 | \( 1 + 2T + 79T^{2} \) |
| 83 | \( 1 - 9iT - 83T^{2} \) |
| 89 | \( 1 - 15T + 89T^{2} \) |
| 97 | \( 1 - 17iT - 97T^{2} \) |
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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
−8.905021246386168361703931587896, −8.086281731106481698194319760636, −7.28821129702796864886004547561, −6.39274022416558980625348178567, −5.54603281014752481032117370050, −4.97543490342114563276405882653, −3.82119010099054735898855585428, −3.36831716625456148757925122775, −2.08417259579842397147968577471, −1.55522204995011653331877589052,
0.21917454267163801647704616208, 1.32958228955180868240647947437, 2.91358797628739838360969115363, 3.57120922888420122699965804783, 4.57651793943974049342545725400, 5.39853921497558192791681881547, 5.80134313608479594715383098324, 7.07543929805163329454165445848, 7.39706131652952899067934576152, 7.86454234592689736798187281105