L(s) = 1 | + (−45 − 67.3i)3-s + 224. i·5-s + 1.75e3·7-s + (−2.51e3 + 6.06e3i)9-s + 6.95e3i·11-s + 2.57e4·13-s + (1.51e4 − 1.01e4i)15-s − 7.48e4i·17-s − 1.89e4·19-s + (−7.87e4 − 1.17e5i)21-s − 4.70e5i·23-s + 3.40e5·25-s + (5.21e5 − 1.03e5i)27-s − 4.60e5i·29-s + 3.51e5·31-s + ⋯ |
L(s) = 1 | + (−0.555 − 0.831i)3-s + 0.359i·5-s + 0.728·7-s + (−0.382 + 0.923i)9-s + 0.475i·11-s + 0.900·13-s + (0.298 − 0.199i)15-s − 0.896i·17-s − 0.145·19-s + (−0.404 − 0.606i)21-s − 1.68i·23-s + 0.870·25-s + (0.980 − 0.195i)27-s − 0.651i·29-s + 0.380·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.555 + 0.831i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.555 + 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{9}{2})\) |
\(\approx\) |
\(1.45789 - 0.779276i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.45789 - 0.779276i\) |
\(L(5)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (45 + 67.3i)T \) |
good | 5 | \( 1 - 224. iT - 3.90e5T^{2} \) |
| 7 | \( 1 - 1.75e3T + 5.76e6T^{2} \) |
| 11 | \( 1 - 6.95e3iT - 2.14e8T^{2} \) |
| 13 | \( 1 - 2.57e4T + 8.15e8T^{2} \) |
| 17 | \( 1 + 7.48e4iT - 6.97e9T^{2} \) |
| 19 | \( 1 + 1.89e4T + 1.69e10T^{2} \) |
| 23 | \( 1 + 4.70e5iT - 7.83e10T^{2} \) |
| 29 | \( 1 + 4.60e5iT - 5.00e11T^{2} \) |
| 31 | \( 1 - 3.51e5T + 8.52e11T^{2} \) |
| 37 | \( 1 - 1.33e6T + 3.51e12T^{2} \) |
| 41 | \( 1 + 1.87e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 - 3.52e6T + 1.16e13T^{2} \) |
| 47 | \( 1 + 4.08e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 - 6.60e6iT - 6.22e13T^{2} \) |
| 59 | \( 1 - 1.37e7iT - 1.46e14T^{2} \) |
| 61 | \( 1 - 7.53e5T + 1.91e14T^{2} \) |
| 67 | \( 1 + 2.26e6T + 4.06e14T^{2} \) |
| 71 | \( 1 - 1.70e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 - 2.76e7T + 8.06e14T^{2} \) |
| 79 | \( 1 - 2.29e7T + 1.51e15T^{2} \) |
| 83 | \( 1 + 4.63e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 + 7.26e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 - 1.47e8T + 7.83e15T^{2} \) |
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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
−13.70260107437466586894635630160, −12.51277768015357918966946082947, −11.43264430915779336999848573295, −10.51076159371345303411999712944, −8.612045762433163807982767093047, −7.36919042493287258435336054758, −6.20578692927325601483921042856, −4.68812500496455518655213585337, −2.40001692032778338434242710382, −0.817325643783789336294085270175,
1.15754116742170621321686701466, 3.61288252650347321324123224000, 4.98065529848868112208831702534, 6.17716237888234916682840836213, 8.172804267250429017577728294865, 9.310256699504618723072463268847, 10.76121746546170846429735256890, 11.46640651235785166298089456403, 12.86274930643841808110116366317, 14.27448520457435243076198718689