L(s) = 1 | + 2·5-s − 4·7-s + 11-s + 13-s − 3·17-s + 19-s − 8·23-s − 25-s + 6·29-s − 8·35-s + 8·37-s − 8·43-s − 2·47-s + 9·49-s + 9·53-s + 2·55-s − 3·59-s + 10·61-s + 2·65-s + 4·67-s + 3·71-s + 4·73-s − 4·77-s + 15·79-s + 9·83-s − 6·85-s − 89-s + ⋯ |
L(s) = 1 | + 0.894·5-s − 1.51·7-s + 0.301·11-s + 0.277·13-s − 0.727·17-s + 0.229·19-s − 1.66·23-s − 1/5·25-s + 1.11·29-s − 1.35·35-s + 1.31·37-s − 1.21·43-s − 0.291·47-s + 9/7·49-s + 1.23·53-s + 0.269·55-s − 0.390·59-s + 1.28·61-s + 0.248·65-s + 0.488·67-s + 0.356·71-s + 0.468·73-s − 0.455·77-s + 1.68·79-s + 0.987·83-s − 0.650·85-s − 0.105·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7524 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7524 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.699641726\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.699641726\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 19 | \( 1 - T \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 15 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 + T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−7.962247417201460406746457206743, −6.91712590397811574462132224251, −6.37874993017178251915084753168, −6.05371720986204754801541260210, −5.21575223828088599635814911243, −4.18206171988797992283948219159, −3.54789942516496121253323654028, −2.62561187310210491070296596963, −1.93662679864400120485049310645, −0.63279094669251850452478475468,
0.63279094669251850452478475468, 1.93662679864400120485049310645, 2.62561187310210491070296596963, 3.54789942516496121253323654028, 4.18206171988797992283948219159, 5.21575223828088599635814911243, 6.05371720986204754801541260210, 6.37874993017178251915084753168, 6.91712590397811574462132224251, 7.962247417201460406746457206743