L(s) = 1 | + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s + (−0.866 + 0.5i)7-s + 0.999i·8-s + (0.499 − 0.866i)14-s + (−0.5 − 0.866i)16-s + (0.866 + 0.5i)23-s + 0.999i·28-s + (0.5 + 0.866i)29-s + (0.866 + 0.499i)32-s + (−0.5 + 0.866i)41-s + (−1.73 + i)43-s − 0.999·46-s + (0.866 − 0.5i)47-s + (−0.5 − 0.866i)56-s + ⋯ |
L(s) = 1 | + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s + (−0.866 + 0.5i)7-s + 0.999i·8-s + (0.499 − 0.866i)14-s + (−0.5 − 0.866i)16-s + (0.866 + 0.5i)23-s + 0.999i·28-s + (0.5 + 0.866i)29-s + (0.866 + 0.499i)32-s + (−0.5 + 0.866i)41-s + (−1.73 + i)43-s − 0.999·46-s + (0.866 − 0.5i)47-s + (−0.5 − 0.866i)56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6117864706\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6117864706\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 - 0.5i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (1.73 - i)T + (0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 + T + T^{2} \) |
| 97 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
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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
−9.109526165950445312348622398351, −8.609175786171695801163369586995, −7.76645004164115085264619163583, −6.89164607494742165251858934610, −6.42687884755821449236328197771, −5.53832768037043317014931884735, −4.82817639706075395559889072690, −3.39485618204379045521709257352, −2.54675585906979557940636393260, −1.26358612528574433633198875633,
0.56670579021120350992513561776, 1.97204088140060324704623947880, 3.03371683191849019474949865838, 3.71583700336393557688251818921, 4.71331755537479533627062239772, 5.97069707077278172145685403557, 6.84216557926792527555072571925, 7.23406170739066301162290690774, 8.301573672966944432143518512501, 8.791044579781959107722314651783