| L(s) = 1 | + 25.8·3-s + 65.5·5-s + 5.43·7-s + 424.·9-s + 234.·11-s − 366.·13-s + 1.69e3·15-s − 9.99e2·17-s − 591.·19-s + 140.·21-s − 529·23-s + 1.17e3·25-s + 4.67e3·27-s + 8.59e3·29-s − 6.92e3·31-s + 6.04e3·33-s + 356.·35-s + 2.83e3·37-s − 9.46e3·39-s + 5.91e3·41-s + 1.24e4·43-s + 2.78e4·45-s + 2.11e4·47-s − 1.67e4·49-s − 2.58e4·51-s − 2.57e4·53-s + 1.53e4·55-s + ⋯ |
| L(s) = 1 | + 1.65·3-s + 1.17·5-s + 0.0419·7-s + 1.74·9-s + 0.583·11-s − 0.601·13-s + 1.94·15-s − 0.838·17-s − 0.375·19-s + 0.0695·21-s − 0.208·23-s + 0.375·25-s + 1.23·27-s + 1.89·29-s − 1.29·31-s + 0.967·33-s + 0.0492·35-s + 0.339·37-s − 0.996·39-s + 0.549·41-s + 1.02·43-s + 2.04·45-s + 1.39·47-s − 0.998·49-s − 1.39·51-s − 1.26·53-s + 0.684·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.728640215\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.728640215\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 23 | \( 1 + 529T \) |
| good | 3 | \( 1 - 25.8T + 243T^{2} \) |
| 5 | \( 1 - 65.5T + 3.12e3T^{2} \) |
| 7 | \( 1 - 5.43T + 1.68e4T^{2} \) |
| 11 | \( 1 - 234.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 366.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 9.99e2T + 1.41e6T^{2} \) |
| 19 | \( 1 + 591.T + 2.47e6T^{2} \) |
| 29 | \( 1 - 8.59e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 6.92e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 2.83e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 5.91e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.24e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.11e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.57e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.55e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.31e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 6.80e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 2.14e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.56e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 2.94e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 6.65e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.67e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 4.61e4T + 8.58e9T^{2} \) |
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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
−13.44680860001552617477386226081, −12.40852988506236465467238797655, −10.58118109281232697222172937597, −9.455688567600143406966295226235, −8.919204969976453099477469916975, −7.61531505706806834743466619665, −6.27902560625897239229379272281, −4.41708796030726615985391047823, −2.79816355862108504622378006332, −1.76128974485690617398339249911,
1.76128974485690617398339249911, 2.79816355862108504622378006332, 4.41708796030726615985391047823, 6.27902560625897239229379272281, 7.61531505706806834743466619665, 8.919204969976453099477469916975, 9.455688567600143406966295226235, 10.58118109281232697222172937597, 12.40852988506236465467238797655, 13.44680860001552617477386226081
