L(s) = 1 | − 5.43i·3-s + 6.12i·5-s + (−6.21 − 3.23i)7-s − 20.5·9-s − 15.2·11-s + 3.00i·13-s + 33.3·15-s − 21.6i·17-s + 11.8i·19-s + (−17.5 + 33.7i)21-s − 1.72·23-s − 12.5·25-s + 62.8i·27-s − 41.1·29-s − 8.50i·31-s + ⋯ |
L(s) = 1 | − 1.81i·3-s + 1.22i·5-s + (−0.887 − 0.461i)7-s − 2.28·9-s − 1.38·11-s + 0.231i·13-s + 2.22·15-s − 1.27i·17-s + 0.626i·19-s + (−0.836 + 1.60i)21-s − 0.0748·23-s − 0.502·25-s + 2.32i·27-s − 1.41·29-s − 0.274i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.887 - 0.461i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.887 - 0.461i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0916327 + 0.374715i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0916327 + 0.374715i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (6.21 + 3.23i)T \) |
good | 3 | \( 1 + 5.43iT - 9T^{2} \) |
| 5 | \( 1 - 6.12iT - 25T^{2} \) |
| 11 | \( 1 + 15.2T + 121T^{2} \) |
| 13 | \( 1 - 3.00iT - 169T^{2} \) |
| 17 | \( 1 + 21.6iT - 289T^{2} \) |
| 19 | \( 1 - 11.8iT - 361T^{2} \) |
| 23 | \( 1 + 1.72T + 529T^{2} \) |
| 29 | \( 1 + 41.1T + 841T^{2} \) |
| 31 | \( 1 + 8.50iT - 961T^{2} \) |
| 37 | \( 1 - 53.1T + 1.36e3T^{2} \) |
| 41 | \( 1 + 43.0iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 36.6T + 1.84e3T^{2} \) |
| 47 | \( 1 + 30.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 30T + 2.80e3T^{2} \) |
| 59 | \( 1 + 73.0iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 5.67iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 28.7T + 4.48e3T^{2} \) |
| 71 | \( 1 + 5.53T + 5.04e3T^{2} \) |
| 73 | \( 1 + 94.9iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 81.5T + 6.24e3T^{2} \) |
| 83 | \( 1 - 86.2iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 27.6iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 100. iT - 9.40e3T^{2} \) |
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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
−11.55221488424310892986176801468, −10.72320346874328132026063416069, −9.568306199105696276595297814482, −7.962471837386000023357054842667, −7.29941607152465910939573877945, −6.64917842554162375374429979650, −5.63075253228860287771920168595, −3.21290243326620881805651351412, −2.28694975553724960250194204790, −0.19377922480347467975089559929,
2.91068982443971425012796922367, 4.18567303099730619734674541181, 5.16595645460576386564824575078, 5.90808386511517854633369669540, 8.084094563895763386293355417443, 8.915420459754204704254412185108, 9.666569995674587410049577517684, 10.42842673464059746078429617340, 11.38990369786516012804410662584, 12.75758036487513779721618521896