L(s) = 1 | + 1.41i·2-s − 2.00·4-s + 8.60i·5-s − 2.82i·8-s − 12.1·10-s − 2.82i·11-s + 12.1·13-s + 4.00·16-s + 25.8i·17-s − 24.3·19-s − 17.2i·20-s + 4.00·22-s + 42.4i·23-s − 49·25-s + 17.2i·26-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 0.500·4-s + 1.72i·5-s − 0.353i·8-s − 1.21·10-s − 0.257i·11-s + 0.935·13-s + 0.250·16-s + 1.51i·17-s − 1.28·19-s − 0.860i·20-s + 0.181·22-s + 1.84i·23-s − 1.95·25-s + 0.661i·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 + 0.577i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.115247611\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.115247611\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 1.41iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 - 8.60iT - 25T^{2} \) |
| 11 | \( 1 + 2.82iT - 121T^{2} \) |
| 13 | \( 1 - 12.1T + 169T^{2} \) |
| 17 | \( 1 - 25.8iT - 289T^{2} \) |
| 19 | \( 1 + 24.3T + 361T^{2} \) |
| 23 | \( 1 - 42.4iT - 529T^{2} \) |
| 29 | \( 1 + 15.5iT - 841T^{2} \) |
| 31 | \( 1 - 24.3T + 961T^{2} \) |
| 37 | \( 1 + 6T + 1.36e3T^{2} \) |
| 41 | \( 1 - 25.8iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 68T + 1.84e3T^{2} \) |
| 47 | \( 1 + 68.8iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 41.0iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 68.8iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 97.3T + 3.72e3T^{2} \) |
| 67 | \( 1 + 104T + 4.48e3T^{2} \) |
| 71 | \( 1 + 70.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 60.8T + 5.32e3T^{2} \) |
| 79 | \( 1 + 20T + 6.24e3T^{2} \) |
| 83 | \( 1 - 6.88e3T^{2} \) |
| 89 | \( 1 - 94.6iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 158.T + 9.40e3T^{2} \) |
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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
−10.43088091220653480470755424316, −9.716211959685369980897113553953, −8.430741829809953871523840979112, −7.941188950066159964315045683271, −6.72879173463548010364093930558, −6.41453302085304345891122172098, −5.53185322348073154281643353134, −3.94592705394470920443596377872, −3.35788222086694448406450937812, −1.87729059707322405065372134023,
0.37708142159957593234561943418, 1.40581170529869928932968647518, 2.69937931496414362329289544349, 4.22986449759016546900166220730, 4.67546311757176193439419532991, 5.66313341273403205750306222062, 6.82329600200919400285136277497, 8.260871567964461576743393530058, 8.639842530703635643149403561826, 9.355933630483436782333172908302