| L(s) = 1 | − 49.4·5-s + 93.1·7-s + 81·9-s + 173.·11-s − 219.·17-s + 361·19-s − 158·23-s + 1.82e3·25-s − 4.60e3·35-s − 800.·43-s − 4.00e3·45-s + 3.07e3·47-s + 6.27e3·49-s − 8.56e3·55-s − 7.41e3·61-s + 7.54e3·63-s + 1.90e3·73-s + 1.61e4·77-s + 6.56e3·81-s − 5.67e3·83-s + 1.08e4·85-s − 1.78e4·95-s + 1.40e4·99-s − 9.99e3·101-s + 7.81e3·115-s − 2.04e4·119-s + ⋯ |
| L(s) = 1 | − 1.97·5-s + 1.90·7-s + 9-s + 1.43·11-s − 0.760·17-s + 19-s − 0.298·23-s + 2.91·25-s − 3.76·35-s − 0.433·43-s − 1.97·45-s + 1.39·47-s + 2.61·49-s − 2.83·55-s − 1.99·61-s + 1.90·63-s + 0.356·73-s + 2.71·77-s + 81-s − 0.824·83-s + 1.50·85-s − 1.97·95-s + 1.43·99-s − 0.980·101-s + 0.591·115-s − 1.44·119-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.601851001\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.601851001\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 - 361T \) |
| good | 3 | \( 1 - 81T^{2} \) |
| 5 | \( 1 + 49.4T + 625T^{2} \) |
| 7 | \( 1 - 93.1T + 2.40e3T^{2} \) |
| 11 | \( 1 - 173.T + 1.46e4T^{2} \) |
| 13 | \( 1 - 2.85e4T^{2} \) |
| 17 | \( 1 + 219.T + 8.35e4T^{2} \) |
| 23 | \( 1 + 158T + 2.79e5T^{2} \) |
| 29 | \( 1 - 7.07e5T^{2} \) |
| 31 | \( 1 - 9.23e5T^{2} \) |
| 37 | \( 1 - 1.87e6T^{2} \) |
| 41 | \( 1 - 2.82e6T^{2} \) |
| 43 | \( 1 + 800.T + 3.41e6T^{2} \) |
| 47 | \( 1 - 3.07e3T + 4.87e6T^{2} \) |
| 53 | \( 1 - 7.89e6T^{2} \) |
| 59 | \( 1 - 1.21e7T^{2} \) |
| 61 | \( 1 + 7.41e3T + 1.38e7T^{2} \) |
| 67 | \( 1 - 2.01e7T^{2} \) |
| 71 | \( 1 - 2.54e7T^{2} \) |
| 73 | \( 1 - 1.90e3T + 2.83e7T^{2} \) |
| 79 | \( 1 - 3.89e7T^{2} \) |
| 83 | \( 1 + 5.67e3T + 4.74e7T^{2} \) |
| 89 | \( 1 - 6.27e7T^{2} \) |
| 97 | \( 1 - 8.85e7T^{2} \) |
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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
−14.01904700270578621902468101096, −12.21847304437388788078895394684, −11.64468282297172972930385943609, −10.85774403611172679270625016559, −8.918084673256899788358037017479, −7.87035126533830974706040219243, −7.09606906488378120870053667561, −4.67789452048106077389420418400, −3.96023389592170013474671415225, −1.22965593474006071425227239528,
1.22965593474006071425227239528, 3.96023389592170013474671415225, 4.67789452048106077389420418400, 7.09606906488378120870053667561, 7.87035126533830974706040219243, 8.918084673256899788358037017479, 10.85774403611172679270625016559, 11.64468282297172972930385943609, 12.21847304437388788078895394684, 14.01904700270578621902468101096
