| L(s) = 1 | − 22.3·3-s + (202. − 591. i)5-s + 1.59e3·7-s − 6.06e3·9-s − 9.15e3i·11-s − 5.55e3i·13-s + (−4.53e3 + 1.32e4i)15-s − 1.04e4i·17-s + 1.21e5i·19-s − 3.57e4·21-s − 2.39e5·23-s + (−3.08e5 − 2.39e5i)25-s + 2.82e5·27-s − 3.01e5·29-s + 7.88e5i·31-s + ⋯ |
| L(s) = 1 | − 0.276·3-s + (0.324 − 0.945i)5-s + 0.665·7-s − 0.923·9-s − 0.625i·11-s − 0.194i·13-s + (−0.0896 + 0.261i)15-s − 0.125i·17-s + 0.929i·19-s − 0.183·21-s − 0.855·23-s + (−0.789 − 0.613i)25-s + 0.531·27-s − 0.426·29-s + 0.853i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.324 - 0.945i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.324 - 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.9749607122\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9749607122\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-202. + 591. i)T \) |
| good | 3 | \( 1 + 22.3T + 6.56e3T^{2} \) |
| 7 | \( 1 - 1.59e3T + 5.76e6T^{2} \) |
| 11 | \( 1 + 9.15e3iT - 2.14e8T^{2} \) |
| 13 | \( 1 + 5.55e3iT - 8.15e8T^{2} \) |
| 17 | \( 1 + 1.04e4iT - 6.97e9T^{2} \) |
| 19 | \( 1 - 1.21e5iT - 1.69e10T^{2} \) |
| 23 | \( 1 + 2.39e5T + 7.83e10T^{2} \) |
| 29 | \( 1 + 3.01e5T + 5.00e11T^{2} \) |
| 31 | \( 1 - 7.88e5iT - 8.52e11T^{2} \) |
| 37 | \( 1 - 1.93e6iT - 3.51e12T^{2} \) |
| 41 | \( 1 + 1.28e6T + 7.98e12T^{2} \) |
| 43 | \( 1 - 6.03e6T + 1.16e13T^{2} \) |
| 47 | \( 1 + 8.00e6T + 2.38e13T^{2} \) |
| 53 | \( 1 - 6.59e6iT - 6.22e13T^{2} \) |
| 59 | \( 1 + 1.67e7iT - 1.46e14T^{2} \) |
| 61 | \( 1 - 5.53e6T + 1.91e14T^{2} \) |
| 67 | \( 1 - 1.45e7T + 4.06e14T^{2} \) |
| 71 | \( 1 - 1.26e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 + 2.28e7iT - 8.06e14T^{2} \) |
| 79 | \( 1 + 3.51e7iT - 1.51e15T^{2} \) |
| 83 | \( 1 + 3.74e7T + 2.25e15T^{2} \) |
| 89 | \( 1 - 2.75e7T + 3.93e15T^{2} \) |
| 97 | \( 1 - 9.97e7iT - 7.83e15T^{2} \) |
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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
−10.48327178471347724525947716614, −9.419760286598986476800007188713, −8.422413155438537695846787615801, −7.949412058843154856794040399344, −6.28543393756762864766570832965, −5.53065436282028530795069782928, −4.71005629358275384512046781292, −3.38519800757701268697012518650, −1.95827921557480579315816410844, −0.910503346637445731914244366492,
0.22863757721920398468773217235, 1.86207552314142110018313999035, 2.71355092701361185972998883317, 4.05130530631474803480281403419, 5.27538853364816544524328561395, 6.16535459434370898227334637605, 7.14833847026536501976421293575, 8.079314416509256309815934094207, 9.221928381707511627331012859512, 10.14947170642241718820777998305
