| L(s) = 1 | − 0.561·5-s − 11-s − 2·13-s − 7.12·17-s − 1.12·19-s − 7.68·23-s − 4.68·25-s + 7.12·29-s + 5.43·31-s + 5.68·37-s + 8.24·41-s − 1.12·43-s − 4·47-s − 7·49-s + 8.24·53-s + 0.561·55-s + 0.315·59-s − 9.36·61-s + 1.12·65-s + 7.68·67-s + 15.6·71-s − 6·73-s + 13.1·79-s + 11.3·83-s + 4·85-s − 0.561·89-s + 0.630·95-s + ⋯ |
| L(s) = 1 | − 0.251·5-s − 0.301·11-s − 0.554·13-s − 1.72·17-s − 0.257·19-s − 1.60·23-s − 0.936·25-s + 1.32·29-s + 0.976·31-s + 0.934·37-s + 1.28·41-s − 0.171·43-s − 0.583·47-s − 49-s + 1.13·53-s + 0.0757·55-s + 0.0410·59-s − 1.19·61-s + 0.139·65-s + 0.938·67-s + 1.86·71-s − 0.702·73-s + 1.47·79-s + 1.24·83-s + 0.433·85-s − 0.0595·89-s + 0.0647·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.211133611\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.211133611\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 5 | \( 1 + 0.561T + 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 + 7.12T + 17T^{2} \) |
| 19 | \( 1 + 1.12T + 19T^{2} \) |
| 23 | \( 1 + 7.68T + 23T^{2} \) |
| 29 | \( 1 - 7.12T + 29T^{2} \) |
| 31 | \( 1 - 5.43T + 31T^{2} \) |
| 37 | \( 1 - 5.68T + 37T^{2} \) |
| 41 | \( 1 - 8.24T + 41T^{2} \) |
| 43 | \( 1 + 1.12T + 43T^{2} \) |
| 47 | \( 1 + 4T + 47T^{2} \) |
| 53 | \( 1 - 8.24T + 53T^{2} \) |
| 59 | \( 1 - 0.315T + 59T^{2} \) |
| 61 | \( 1 + 9.36T + 61T^{2} \) |
| 67 | \( 1 - 7.68T + 67T^{2} \) |
| 71 | \( 1 - 15.6T + 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 - 13.1T + 79T^{2} \) |
| 83 | \( 1 - 11.3T + 83T^{2} \) |
| 89 | \( 1 + 0.561T + 89T^{2} \) |
| 97 | \( 1 - 5.68T + 97T^{2} \) |
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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
−8.019430595467747593397337984093, −7.46803125053773855341117261493, −6.33428244849035194162063066223, −6.27417661180688312398336058062, −4.98215324249077053170380155304, −4.45555337512782494579671955400, −3.74790363952911656594424461630, −2.56693275154179492329905877024, −2.06308837701462828438497134662, −0.54421452148532331857859714146,
0.54421452148532331857859714146, 2.06308837701462828438497134662, 2.56693275154179492329905877024, 3.74790363952911656594424461630, 4.45555337512782494579671955400, 4.98215324249077053170380155304, 6.27417661180688312398336058062, 6.33428244849035194162063066223, 7.46803125053773855341117261493, 8.019430595467747593397337984093
