| L(s) = 1 | − 2i·7-s + 2i·13-s − 3i·17-s − 5·19-s − 3i·23-s − 6·29-s + 5·31-s − 2i·37-s − 12·41-s + 8i·43-s − 12i·47-s + 3·49-s + 3i·53-s + 6·59-s − 7·61-s + ⋯ |
| L(s) = 1 | − 0.755i·7-s + 0.554i·13-s − 0.727i·17-s − 1.14·19-s − 0.625i·23-s − 1.11·29-s + 0.898·31-s − 0.328i·37-s − 1.87·41-s + 1.21i·43-s − 1.75i·47-s + 0.428·49-s + 0.412i·53-s + 0.781·59-s − 0.896·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6298786856\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6298786856\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 + 3iT - 17T^{2} \) |
| 19 | \( 1 + 5T + 19T^{2} \) |
| 23 | \( 1 + 3iT - 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 - 5T + 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 + 12T + 41T^{2} \) |
| 43 | \( 1 - 8iT - 43T^{2} \) |
| 47 | \( 1 + 12iT - 47T^{2} \) |
| 53 | \( 1 - 3iT - 53T^{2} \) |
| 59 | \( 1 - 6T + 59T^{2} \) |
| 61 | \( 1 + 7T + 61T^{2} \) |
| 67 | \( 1 + 2iT - 67T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 + 16iT - 73T^{2} \) |
| 79 | \( 1 - T + 79T^{2} \) |
| 83 | \( 1 - 15iT - 83T^{2} \) |
| 89 | \( 1 + 12T + 89T^{2} \) |
| 97 | \( 1 - 16iT - 97T^{2} \) |
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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
−8.533760035414045311405967542191, −7.74070404352300971470179119877, −6.90644493267818336761274269203, −6.43073832395284345494871849228, −5.32564962376082343222952214163, −4.47530553601358451172067260392, −3.81462926068523236908028476419, −2.69345992915804009536340163311, −1.61374302097108356936335356159, −0.19522241933984479245536358872,
1.54933301357522248129436025119, 2.53406835853089939199932237955, 3.51657820152373293147002787362, 4.43171207048348375439391797315, 5.43131183881456093268905833899, 5.99494501538510585038114112460, 6.83193214428123948603486036627, 7.74900217606276847822930402353, 8.507701003224320979849503734115, 8.965139625393844551446875413928
