L(s) = 1 | + 81i·3-s + 1.42e3i·5-s + 6.38e3·7-s − 6.56e3·9-s − 4.07e3i·11-s + 1.54e5i·13-s − 1.15e5·15-s + 2.55e5·17-s − 1.01e6i·19-s + 5.16e5i·21-s − 1.56e6·23-s − 8.63e4·25-s − 5.31e5i·27-s − 7.26e6i·29-s − 6.11e5·31-s + ⋯ |
L(s) = 1 | + 0.577i·3-s + 1.02i·5-s + 1.00·7-s − 0.333·9-s − 0.0838i·11-s + 1.49i·13-s − 0.589·15-s + 0.742·17-s − 1.79i·19-s + 0.580i·21-s − 1.16·23-s − 0.0442·25-s − 0.192i·27-s − 1.90i·29-s − 0.119·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(5)\) |
\(\approx\) |
\(1.553481709\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.553481709\) |
\(L(\frac{11}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - 81iT \) |
good | 5 | \( 1 - 1.42e3iT - 1.95e6T^{2} \) |
| 7 | \( 1 - 6.38e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 4.07e3iT - 2.35e9T^{2} \) |
| 13 | \( 1 - 1.54e5iT - 1.06e10T^{2} \) |
| 17 | \( 1 - 2.55e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 1.01e6iT - 3.22e11T^{2} \) |
| 23 | \( 1 + 1.56e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 7.26e6iT - 1.45e13T^{2} \) |
| 31 | \( 1 + 6.11e5T + 2.64e13T^{2} \) |
| 37 | \( 1 - 4.64e5iT - 1.29e14T^{2} \) |
| 41 | \( 1 - 2.04e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 2.95e7iT - 5.02e14T^{2} \) |
| 47 | \( 1 - 1.31e6T + 1.11e15T^{2} \) |
| 53 | \( 1 + 5.76e7iT - 3.29e15T^{2} \) |
| 59 | \( 1 + 6.35e7iT - 8.66e15T^{2} \) |
| 61 | \( 1 - 5.01e7iT - 1.16e16T^{2} \) |
| 67 | \( 1 + 2.90e8iT - 2.72e16T^{2} \) |
| 71 | \( 1 + 3.18e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 3.49e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 4.29e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 3.40e8iT - 1.86e17T^{2} \) |
| 89 | \( 1 - 5.24e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 9.61e8T + 7.60e17T^{2} \) |
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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
−9.755580221687614544707739322857, −8.890478187381416279297326731886, −7.82118547702766743579109774622, −6.92310779384410871818945153124, −5.95013227780702922922768655085, −4.72768992009079690473607078486, −3.99576460012520932920175991025, −2.72314116443819701668315608989, −1.84755788523286563154487448434, −0.28033205758775022062158513425,
1.13410334799128374293977990323, 1.47443600822487002897145608814, 2.96798691907025031669734554047, 4.24307591277655742724695494639, 5.36863062837891679248077033515, 5.86533929774821676769163087088, 7.52025829276237178316769768231, 8.044047936888361224117968905427, 8.716431774847270578924255260333, 9.988940545557056081805541790070