| L(s) = 1 | − 16.7i·3-s + 94.1i·7-s − 36.4·9-s + 143.·11-s − 421. i·13-s + 1.98e3i·17-s + 1.31e3·19-s + 1.57e3·21-s + 4.02e3i·23-s − 3.45e3i·27-s + 6.41e3·29-s − 2.35e3·31-s − 2.40e3i·33-s − 7.87e3i·37-s − 7.04e3·39-s + ⋯ |
| L(s) = 1 | − 1.07i·3-s + 0.726i·7-s − 0.149·9-s + 0.358·11-s − 0.691i·13-s + 1.66i·17-s + 0.837·19-s + 0.778·21-s + 1.58i·23-s − 0.911i·27-s + 1.41·29-s − 0.439·31-s − 0.383i·33-s − 0.945i·37-s − 0.741·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.130285273\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.130285273\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 16.7iT - 243T^{2} \) |
| 7 | \( 1 - 94.1iT - 1.68e4T^{2} \) |
| 11 | \( 1 - 143.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 421. iT - 3.71e5T^{2} \) |
| 17 | \( 1 - 1.98e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 1.31e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 4.02e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 - 6.41e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 2.35e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 7.87e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 1.50e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.14e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 2.15e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 + 9.56e3iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 4.27e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.21e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.03e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 3.60e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 6.34e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 - 8.99e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 3.82e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 + 5.74e3T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.78e5iT - 8.58e9T^{2} \) |
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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
−11.81553963276851196392608768910, −10.60489758655335525889702317328, −9.429828878448148283312591022087, −8.288077925443334931152754554731, −7.48236443857403381000736574194, −6.33107349795434463400165443388, −5.44062763867067363440672798555, −3.66963154977059156917830317509, −2.15199276393246822627597818817, −1.01336140216721744538601795209,
0.879002499092536039593285989187, 2.89989297697243845827381778246, 4.25264503609791318396816609494, 4.88442844167355178674293189297, 6.52832107721148282458173637386, 7.52987174481569512291311133334, 8.994884414231492587133367352895, 9.697989549006947280771956143919, 10.57276691446487375312223216508, 11.45641089003378172826452328440
