| L(s) = 1 | + 2i·3-s − 2i·7-s − 9-s − 4·11-s + 6i·13-s + 2i·17-s − 8·19-s + 4·21-s + 6i·23-s + 4i·27-s + 2·29-s + 4·31-s − 8i·33-s + 2i·37-s − 12·39-s + ⋯ |
| L(s) = 1 | + 1.15i·3-s − 0.755i·7-s − 0.333·9-s − 1.20·11-s + 1.66i·13-s + 0.485i·17-s − 1.83·19-s + 0.872·21-s + 1.25i·23-s + 0.769i·27-s + 0.371·29-s + 0.718·31-s − 1.39i·33-s + 0.328i·37-s − 1.92·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.218802 + 0.926860i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.218802 + 0.926860i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 2iT - 3T^{2} \) |
| 7 | \( 1 + 2iT - 7T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 - 6iT - 13T^{2} \) |
| 17 | \( 1 - 2iT - 17T^{2} \) |
| 19 | \( 1 + 8T + 19T^{2} \) |
| 23 | \( 1 - 6iT - 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 + 10T + 41T^{2} \) |
| 43 | \( 1 - 2iT - 43T^{2} \) |
| 47 | \( 1 + 2iT - 47T^{2} \) |
| 53 | \( 1 + 2iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + 6iT - 67T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 + 10iT - 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 - 10iT - 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 - 10iT - 97T^{2} \) |
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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
−10.47491908197939904203647127102, −9.985444267125944034037253921248, −9.054619824260205469458065305317, −8.232494828028282268116699260376, −7.15486157267252801260739776920, −6.26634532657254455606529633165, −4.94758215657426926633425028408, −4.34550743996633793885571959651, −3.49911211162865093505811948464, −1.94460540979308359566598400772,
0.44797271786889384386884608863, 2.19766779638084244796385281511, 2.88305794249530315726189983479, 4.63324212096648958181487935239, 5.64743794391825044208365965319, 6.41993009569643865772210702014, 7.37486174029341564514622938239, 8.255759889897118346353240409922, 8.610622263332691695908064414208, 10.21298246523348427992405062600
