| L(s) = 1 | − 5-s − 4·11-s − 0.372·13-s + 6.74·17-s + 4·19-s − 6.74·23-s + 25-s − 8.37·29-s + 4.74·31-s + 37-s + 4.74·41-s + 1.62·43-s + 7.11·47-s − 7·49-s + 1.62·53-s + 4·55-s + 0.372·59-s − 6.74·61-s + 0.372·65-s − 4·67-s − 4.74·71-s − 2.74·73-s + 12.7·79-s − 3.11·83-s − 6.74·85-s − 17.1·89-s − 4·95-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.20·11-s − 0.103·13-s + 1.63·17-s + 0.917·19-s − 1.40·23-s + 0.200·25-s − 1.55·29-s + 0.852·31-s + 0.164·37-s + 0.740·41-s + 0.248·43-s + 1.03·47-s − 49-s + 0.223·53-s + 0.539·55-s + 0.0484·59-s − 0.863·61-s + 0.0461·65-s − 0.488·67-s − 0.563·71-s − 0.321·73-s + 1.43·79-s − 0.342·83-s − 0.731·85-s − 1.81·89-s − 0.410·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6660 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6660 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 37 | \( 1 - T \) |
| good | 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 + 0.372T + 13T^{2} \) |
| 17 | \( 1 - 6.74T + 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 + 6.74T + 23T^{2} \) |
| 29 | \( 1 + 8.37T + 29T^{2} \) |
| 31 | \( 1 - 4.74T + 31T^{2} \) |
| 41 | \( 1 - 4.74T + 41T^{2} \) |
| 43 | \( 1 - 1.62T + 43T^{2} \) |
| 47 | \( 1 - 7.11T + 47T^{2} \) |
| 53 | \( 1 - 1.62T + 53T^{2} \) |
| 59 | \( 1 - 0.372T + 59T^{2} \) |
| 61 | \( 1 + 6.74T + 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 + 4.74T + 71T^{2} \) |
| 73 | \( 1 + 2.74T + 73T^{2} \) |
| 79 | \( 1 - 12.7T + 79T^{2} \) |
| 83 | \( 1 + 3.11T + 83T^{2} \) |
| 89 | \( 1 + 17.1T + 89T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
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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
−7.75795107131229239908633460362, −7.22467227644050315026492265280, −6.00007348626062261496166314916, −5.60145796700273914710383545892, −4.82589889099145541557658433674, −3.91819000809732456518493564607, −3.20234896084516432008657866776, −2.40303247958024036503573884146, −1.22409040604588009083980196602, 0,
1.22409040604588009083980196602, 2.40303247958024036503573884146, 3.20234896084516432008657866776, 3.91819000809732456518493564607, 4.82589889099145541557658433674, 5.60145796700273914710383545892, 6.00007348626062261496166314916, 7.22467227644050315026492265280, 7.75795107131229239908633460362
