| L(s) = 1 | − 76.7·3-s − 615.·5-s − 255.·7-s − 1.37e4·9-s + 3.55e4·11-s + 2.86e4·13-s + 4.72e4·15-s + 1.35e5·17-s + 9.45e5·19-s + 1.95e4·21-s + 1.79e6·23-s − 1.57e6·25-s + 2.56e6·27-s + 2.03e6·29-s + 5.52e6·31-s − 2.73e6·33-s + 1.57e5·35-s + 1.36e7·37-s − 2.19e6·39-s − 1.77e7·41-s − 2.06e7·43-s + 8.49e6·45-s − 5.44e7·47-s − 4.02e7·49-s − 1.03e7·51-s + 7.15e7·53-s − 2.19e7·55-s + ⋯ |
| L(s) = 1 | − 0.546·3-s − 0.440·5-s − 0.0401·7-s − 0.700·9-s + 0.732·11-s + 0.277·13-s + 0.241·15-s + 0.393·17-s + 1.66·19-s + 0.0219·21-s + 1.33·23-s − 0.805·25-s + 0.930·27-s + 0.533·29-s + 1.07·31-s − 0.400·33-s + 0.0177·35-s + 1.19·37-s − 0.151·39-s − 0.980·41-s − 0.920·43-s + 0.308·45-s − 1.62·47-s − 0.998·49-s − 0.215·51-s + 1.24·53-s − 0.322·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.359152305\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.359152305\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 + 76.7T + 1.96e4T^{2} \) |
| 5 | \( 1 + 615.T + 1.95e6T^{2} \) |
| 7 | \( 1 + 255.T + 4.03e7T^{2} \) |
| 11 | \( 1 - 3.55e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 2.86e4T + 1.06e10T^{2} \) |
| 17 | \( 1 - 1.35e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 9.45e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 1.79e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 2.03e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 5.52e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 1.36e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + 1.77e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 2.06e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 5.44e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 7.15e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 5.63e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 1.23e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 1.22e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 3.18e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 1.63e7T + 5.88e16T^{2} \) |
| 79 | \( 1 - 2.03e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 2.65e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 5.49e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 7.33e8T + 7.60e17T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.79305222290413780289794973323, −13.57567913480805102164165110285, −11.94452044378647990393479238197, −11.32917421250859739623655999375, −9.681526699589547275206057891447, −8.181821651124924961209727073831, −6.58852391707297878005497697240, −5.12481652347899164164823731595, −3.28459600820393656966544583936, −0.891075367608224006255621262778,
0.891075367608224006255621262778, 3.28459600820393656966544583936, 5.12481652347899164164823731595, 6.58852391707297878005497697240, 8.181821651124924961209727073831, 9.681526699589547275206057891447, 11.32917421250859739623655999375, 11.94452044378647990393479238197, 13.57567913480805102164165110285, 14.79305222290413780289794973323
