| L(s) = 1 | − 18.6·3-s + 49·7-s + 106.·9-s + 259.·11-s + 1.12e3·13-s + 459.·17-s + 1.33e3·19-s − 916.·21-s − 1.35e3·23-s + 2.55e3·27-s + 867.·29-s + 5.07e3·31-s − 4.85e3·33-s − 1.01e4·37-s − 2.09e4·39-s + 1.49e4·41-s − 9.68e3·43-s + 4.29e3·47-s + 2.40e3·49-s − 8.59e3·51-s − 2.82e4·53-s − 2.49e4·57-s − 1.36e4·59-s + 2.58e3·61-s + 5.22e3·63-s + 3.52e4·67-s + 2.53e4·69-s + ⋯ |
| L(s) = 1 | − 1.19·3-s + 0.377·7-s + 0.438·9-s + 0.646·11-s + 1.83·13-s + 0.385·17-s + 0.848·19-s − 0.453·21-s − 0.533·23-s + 0.673·27-s + 0.191·29-s + 0.947·31-s − 0.775·33-s − 1.22·37-s − 2.20·39-s + 1.38·41-s − 0.799·43-s + 0.283·47-s + 0.142·49-s − 0.462·51-s − 1.38·53-s − 1.01·57-s − 0.509·59-s + 0.0890·61-s + 0.165·63-s + 0.959·67-s + 0.640·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.757338529\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.757338529\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - 49T \) |
| good | 3 | \( 1 + 18.6T + 243T^{2} \) |
| 11 | \( 1 - 259.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 1.12e3T + 3.71e5T^{2} \) |
| 17 | \( 1 - 459.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.33e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 1.35e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 867.T + 2.05e7T^{2} \) |
| 31 | \( 1 - 5.07e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.01e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.49e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 9.68e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 4.29e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.82e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.36e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.58e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.52e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 5.15e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.62e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 2.06e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 8.32e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 2.14e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.75e5T + 8.58e9T^{2} \) |
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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
−9.842648800205314734809179339662, −8.777805430436396960559950571680, −7.992858216347186459970025812910, −6.75553549785191597257535995572, −6.07711436117177291969375464234, −5.36448493887808894179724537400, −4.29606901027277212777221561598, −3.26113642265491139414978452250, −1.51013855104589983170821198781, −0.73294988563757686604383000782,
0.73294988563757686604383000782, 1.51013855104589983170821198781, 3.26113642265491139414978452250, 4.29606901027277212777221561598, 5.36448493887808894179724537400, 6.07711436117177291969375464234, 6.75553549785191597257535995572, 7.992858216347186459970025812910, 8.777805430436396960559950571680, 9.842648800205314734809179339662
