| L(s) = 1 | + 141.·3-s + 3.12e3·5-s − 8.55e4·7-s − 1.57e5·9-s − 7.67e5·11-s + 2.20e5·13-s + 4.42e5·15-s − 9.30e5·17-s + 1.77e7·19-s − 1.21e7·21-s + 3.99e7·23-s + 9.76e6·25-s − 4.73e7·27-s + 7.68e7·29-s + 2.96e7·31-s − 1.08e8·33-s − 2.67e8·35-s + 5.40e7·37-s + 3.13e7·39-s + 1.26e8·41-s + 2.88e8·43-s − 4.90e8·45-s + 1.57e9·47-s + 5.34e9·49-s − 1.31e8·51-s − 4.09e9·53-s − 2.39e9·55-s + ⋯ |
| L(s) = 1 | + 0.336·3-s + 0.447·5-s − 1.92·7-s − 0.886·9-s − 1.43·11-s + 0.165·13-s + 0.150·15-s − 0.158·17-s + 1.64·19-s − 0.648·21-s + 1.29·23-s + 0.199·25-s − 0.635·27-s + 0.695·29-s + 0.185·31-s − 0.483·33-s − 0.860·35-s + 0.128·37-s + 0.0555·39-s + 0.169·41-s + 0.299·43-s − 0.396·45-s + 0.998·47-s + 2.70·49-s − 0.0535·51-s − 1.34·53-s − 0.642·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(1.401862016\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.401862016\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - 3.12e3T \) |
| good | 3 | \( 1 - 141.T + 1.77e5T^{2} \) |
| 7 | \( 1 + 8.55e4T + 1.97e9T^{2} \) |
| 11 | \( 1 + 7.67e5T + 2.85e11T^{2} \) |
| 13 | \( 1 - 2.20e5T + 1.79e12T^{2} \) |
| 17 | \( 1 + 9.30e5T + 3.42e13T^{2} \) |
| 19 | \( 1 - 1.77e7T + 1.16e14T^{2} \) |
| 23 | \( 1 - 3.99e7T + 9.52e14T^{2} \) |
| 29 | \( 1 - 7.68e7T + 1.22e16T^{2} \) |
| 31 | \( 1 - 2.96e7T + 2.54e16T^{2} \) |
| 37 | \( 1 - 5.40e7T + 1.77e17T^{2} \) |
| 41 | \( 1 - 1.26e8T + 5.50e17T^{2} \) |
| 43 | \( 1 - 2.88e8T + 9.29e17T^{2} \) |
| 47 | \( 1 - 1.57e9T + 2.47e18T^{2} \) |
| 53 | \( 1 + 4.09e9T + 9.26e18T^{2} \) |
| 59 | \( 1 + 3.77e9T + 3.01e19T^{2} \) |
| 61 | \( 1 + 9.64e9T + 4.35e19T^{2} \) |
| 67 | \( 1 - 1.63e10T + 1.22e20T^{2} \) |
| 71 | \( 1 + 1.03e10T + 2.31e20T^{2} \) |
| 73 | \( 1 - 4.27e9T + 3.13e20T^{2} \) |
| 79 | \( 1 - 1.96e10T + 7.47e20T^{2} \) |
| 83 | \( 1 - 1.35e10T + 1.28e21T^{2} \) |
| 89 | \( 1 - 2.25e10T + 2.77e21T^{2} \) |
| 97 | \( 1 + 1.08e11T + 7.15e21T^{2} \) |
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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
−12.37610238729137597377186479518, −10.85173526004524126222706135784, −9.778468605058932626147433310899, −8.998833867039948831288729870865, −7.55339926950345025399357872459, −6.27890497429824962443715758848, −5.27029237402322867639868407842, −3.19810684702355697095109601902, −2.71689032704121873084413907535, −0.59425195306094299501342264143,
0.59425195306094299501342264143, 2.71689032704121873084413907535, 3.19810684702355697095109601902, 5.27029237402322867639868407842, 6.27890497429824962443715758848, 7.55339926950345025399357872459, 8.998833867039948831288729870865, 9.778468605058932626147433310899, 10.85173526004524126222706135784, 12.37610238729137597377186479518
