L(s) = 1 | − 9.93e3·5-s + 1.68e4·7-s − 7.28e5·11-s + 7.43e5·13-s − 3.15e6·17-s + 1.12e7·19-s − 9.99e5·23-s + 4.98e7·25-s + 1.65e8·29-s − 2.48e8·31-s − 1.66e8·35-s + 4.28e8·37-s + 8.47e8·41-s + 2.27e8·43-s − 1.26e9·47-s + 2.82e8·49-s − 5.00e9·53-s + 7.23e9·55-s + 7.66e9·59-s + 1.03e10·61-s − 7.38e9·65-s − 4.42e9·67-s + 1.58e10·71-s + 2.59e10·73-s − 1.22e10·77-s + 1.54e9·79-s − 3.30e10·83-s + ⋯ |
L(s) = 1 | − 1.42·5-s + 0.377·7-s − 1.36·11-s + 0.555·13-s − 0.539·17-s + 1.03·19-s − 0.0323·23-s + 1.02·25-s + 1.49·29-s − 1.55·31-s − 0.537·35-s + 1.01·37-s + 1.14·41-s + 0.236·43-s − 0.804·47-s + 0.142·49-s − 1.64·53-s + 1.93·55-s + 1.39·59-s + 1.56·61-s − 0.789·65-s − 0.400·67-s + 1.04·71-s + 1.46·73-s − 0.515·77-s + 0.0563·79-s − 0.921·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(6)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{13}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - 1.68e4T \) |
good | 5 | \( 1 + 9.93e3T + 4.88e7T^{2} \) |
| 11 | \( 1 + 7.28e5T + 2.85e11T^{2} \) |
| 13 | \( 1 - 7.43e5T + 1.79e12T^{2} \) |
| 17 | \( 1 + 3.15e6T + 3.42e13T^{2} \) |
| 19 | \( 1 - 1.12e7T + 1.16e14T^{2} \) |
| 23 | \( 1 + 9.99e5T + 9.52e14T^{2} \) |
| 29 | \( 1 - 1.65e8T + 1.22e16T^{2} \) |
| 31 | \( 1 + 2.48e8T + 2.54e16T^{2} \) |
| 37 | \( 1 - 4.28e8T + 1.77e17T^{2} \) |
| 41 | \( 1 - 8.47e8T + 5.50e17T^{2} \) |
| 43 | \( 1 - 2.27e8T + 9.29e17T^{2} \) |
| 47 | \( 1 + 1.26e9T + 2.47e18T^{2} \) |
| 53 | \( 1 + 5.00e9T + 9.26e18T^{2} \) |
| 59 | \( 1 - 7.66e9T + 3.01e19T^{2} \) |
| 61 | \( 1 - 1.03e10T + 4.35e19T^{2} \) |
| 67 | \( 1 + 4.42e9T + 1.22e20T^{2} \) |
| 71 | \( 1 - 1.58e10T + 2.31e20T^{2} \) |
| 73 | \( 1 - 2.59e10T + 3.13e20T^{2} \) |
| 79 | \( 1 - 1.54e9T + 7.47e20T^{2} \) |
| 83 | \( 1 + 3.30e10T + 1.28e21T^{2} \) |
| 89 | \( 1 + 4.41e9T + 2.77e21T^{2} \) |
| 97 | \( 1 + 4.12e10T + 7.15e21T^{2} \) |
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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
−9.626470359504987201357301970055, −8.319443097808592607909403348666, −7.87839131160116467919038725656, −6.92836636642676057420836856410, −5.49376245667337452807366594688, −4.54426810963010913156716670786, −3.55434205700489161658954549588, −2.51257861866304224778746375325, −0.974688311950984533678453465735, 0,
0.974688311950984533678453465735, 2.51257861866304224778746375325, 3.55434205700489161658954549588, 4.54426810963010913156716670786, 5.49376245667337452807366594688, 6.92836636642676057420836856410, 7.87839131160116467919038725656, 8.319443097808592607909403348666, 9.626470359504987201357301970055