L(s) = 1 | + (28.8 + 13.8i)2-s − 196.·3-s + (641. + 798. i)4-s + 5.38e3i·5-s + (−5.67e3 − 2.71e3i)6-s − 6.61e3i·7-s + (7.47e3 + 3.19e4i)8-s − 2.03e4·9-s + (−7.44e4 + 1.55e5i)10-s + 2.21e5·11-s + (−1.26e5 − 1.56e5i)12-s − 3.61e5i·13-s + (9.14e4 − 1.90e5i)14-s − 1.05e6i·15-s + (−2.25e5 + 1.02e6i)16-s + 7.76e5·17-s + ⋯ |
L(s) = 1 | + (0.901 + 0.432i)2-s − 0.809·3-s + (0.626 + 0.779i)4-s + 1.72i·5-s + (−0.729 − 0.349i)6-s − 0.393i·7-s + (0.228 + 0.973i)8-s − 0.344·9-s + (−0.744 + 1.55i)10-s + 1.37·11-s + (−0.507 − 0.630i)12-s − 0.974i·13-s + (0.170 − 0.354i)14-s − 1.39i·15-s + (−0.215 + 0.976i)16-s + 0.546·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.228 - 0.973i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.228 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{11}{2})\) |
\(\approx\) |
\(1.20762 + 1.52323i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.20762 + 1.52323i\) |
\(L(6)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-28.8 - 13.8i)T \) |
good | 3 | \( 1 + 196.T + 5.90e4T^{2} \) |
| 5 | \( 1 - 5.38e3iT - 9.76e6T^{2} \) |
| 7 | \( 1 + 6.61e3iT - 2.82e8T^{2} \) |
| 11 | \( 1 - 2.21e5T + 2.59e10T^{2} \) |
| 13 | \( 1 + 3.61e5iT - 1.37e11T^{2} \) |
| 17 | \( 1 - 7.76e5T + 2.01e12T^{2} \) |
| 19 | \( 1 - 1.46e6T + 6.13e12T^{2} \) |
| 23 | \( 1 - 2.34e6iT - 4.14e13T^{2} \) |
| 29 | \( 1 - 2.88e7iT - 4.20e14T^{2} \) |
| 31 | \( 1 + 3.00e7iT - 8.19e14T^{2} \) |
| 37 | \( 1 + 1.16e7iT - 4.80e15T^{2} \) |
| 41 | \( 1 - 4.70e7T + 1.34e16T^{2} \) |
| 43 | \( 1 - 3.52e7T + 2.16e16T^{2} \) |
| 47 | \( 1 + 4.23e8iT - 5.25e16T^{2} \) |
| 53 | \( 1 - 9.16e6iT - 1.74e17T^{2} \) |
| 59 | \( 1 + 1.03e9T + 5.11e17T^{2} \) |
| 61 | \( 1 - 1.37e8iT - 7.13e17T^{2} \) |
| 67 | \( 1 + 7.94e8T + 1.82e18T^{2} \) |
| 71 | \( 1 + 2.04e9iT - 3.25e18T^{2} \) |
| 73 | \( 1 + 1.88e9T + 4.29e18T^{2} \) |
| 79 | \( 1 - 1.04e9iT - 9.46e18T^{2} \) |
| 83 | \( 1 - 3.77e9T + 1.55e19T^{2} \) |
| 89 | \( 1 + 4.07e9T + 3.11e19T^{2} \) |
| 97 | \( 1 - 1.76e9T + 7.37e19T^{2} \) |
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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
−19.93158000937555625262494662400, −17.98134864088245443465164783460, −16.80707156048583088380150451527, −15.00317075864064807774279670860, −14.02278637976655651086662161876, −11.86746219357131094485037951801, −10.74886799677693933921105097839, −7.20427641006092238996840351375, −5.91629274648150166878305522810, −3.33854134247623960022608252835,
1.14962116667345822782600200498, 4.52544855981285131820693375679, 5.96685955414978732359088055394, 9.221588757400835911720773657783, 11.67173056203627760558484140625, 12.33254132398882362400657116391, 14.08466437818703087934857438695, 16.13555455165419127713244546709, 17.09488010200734778783735113807, 19.42018154588921149618958348666