L(s) = 1 | + 3i·7-s + 5·11-s − 4i·13-s − 8i·17-s − 2·19-s − 2i·23-s − 6·29-s − 7·31-s − 6i·37-s − 6·41-s + 2i·43-s + 6i·47-s − 2·49-s − 5i·53-s + 4·59-s + ⋯ |
L(s) = 1 | + 1.13i·7-s + 1.50·11-s − 1.10i·13-s − 1.94i·17-s − 0.458·19-s − 0.417i·23-s − 1.11·29-s − 1.25·31-s − 0.986i·37-s − 0.937·41-s + 0.304i·43-s + 0.875i·47-s − 0.285·49-s − 0.686i·53-s + 0.520·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.076106431\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.076106431\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 3iT - 7T^{2} \) |
| 11 | \( 1 - 5T + 11T^{2} \) |
| 13 | \( 1 + 4iT - 13T^{2} \) |
| 17 | \( 1 + 8iT - 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 + 2iT - 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + 7T + 31T^{2} \) |
| 37 | \( 1 + 6iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - 2iT - 43T^{2} \) |
| 47 | \( 1 - 6iT - 47T^{2} \) |
| 53 | \( 1 + 5iT - 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 + 8T + 61T^{2} \) |
| 67 | \( 1 + 10iT - 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 + iT - 73T^{2} \) |
| 79 | \( 1 + 16T + 79T^{2} \) |
| 83 | \( 1 - 11iT - 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 + iT - 97T^{2} \) |
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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
−7.920959637946580340140411278262, −7.21787666304516195158655795546, −6.50783527736378785509248610138, −5.67854238736469912943297810963, −5.21520433357114184351626430047, −4.23797212329524637497636238290, −3.32617298361677094913514036588, −2.58043159623514795103542932919, −1.60253438084587106743806142977, −0.27195242167192809556675305017,
1.41053815764737135214798380682, 1.80504140097477334791153832179, 3.47778947356838967625892409343, 3.98694790468124806575689375957, 4.41420557126733876977935607197, 5.68238009373076973321178143258, 6.36206863660346596386242591381, 6.98979661228171531634185494352, 7.50564367765200221301608759911, 8.615495351766044192084741097264