L(s) = 1 | − i·7-s + 3·9-s − 4·11-s − 6i·13-s − 2i·17-s − 6·29-s − 8·31-s + 10i·37-s + 2·41-s − 4i·43-s + 8i·47-s − 49-s − 2i·53-s − 8·59-s − 14·61-s + ⋯ |
L(s) = 1 | − 0.377i·7-s + 9-s − 1.20·11-s − 1.66i·13-s − 0.485i·17-s − 1.11·29-s − 1.43·31-s + 1.64i·37-s + 0.312·41-s − 0.609i·43-s + 1.16i·47-s − 0.142·49-s − 0.274i·53-s − 1.04·59-s − 1.79·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7464900134\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7464900134\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + iT \) |
good | 3 | \( 1 - 3T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 + 6iT - 13T^{2} \) |
| 17 | \( 1 + 2iT - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 - 10iT - 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 - 8iT - 47T^{2} \) |
| 53 | \( 1 + 2iT - 53T^{2} \) |
| 59 | \( 1 + 8T + 59T^{2} \) |
| 61 | \( 1 + 14T + 61T^{2} \) |
| 67 | \( 1 + 12iT - 67T^{2} \) |
| 71 | \( 1 - 16T + 71T^{2} \) |
| 73 | \( 1 - 2iT - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 + 8iT - 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 + 2iT - 97T^{2} \) |
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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
−8.265899950052732115419759938554, −7.63705964918998725770461433039, −7.24858340210446861545299137651, −6.11208397826544489742084323069, −5.29690891153501965584474231139, −4.67549331470859371004058129327, −3.55905166563266995484188735299, −2.80724664657204643598495483787, −1.55284920240425145576299385101, −0.22725873497587523262646158805,
1.64769041325788342416736408368, 2.33132608666228447039110626194, 3.68288765754575919343428686690, 4.35358457679376603801551612536, 5.27684196099387196231529700968, 6.01725356963673534933512591456, 7.04534924918139999345931367787, 7.45560266832340850254893034282, 8.377609988003526667801267576654, 9.295547717678141579623653251891