L(s) = 1 | + (−1.57 + 2.55i)3-s − 1.31i·5-s + 10.2·7-s + (−4.01 − 8.05i)9-s − 16.6i·11-s − 18.7·13-s + (3.35 + 2.07i)15-s − 4.38i·17-s + 11.5·19-s + (−16.1 + 26.1i)21-s − 16.7i·23-s + 23.2·25-s + (26.8 + 2.46i)27-s − 12.5i·29-s + 20.3·31-s + ⋯ |
L(s) = 1 | + (−0.526 + 0.850i)3-s − 0.263i·5-s + 1.46·7-s + (−0.446 − 0.894i)9-s − 1.51i·11-s − 1.44·13-s + (0.223 + 0.138i)15-s − 0.257i·17-s + 0.608·19-s + (−0.769 + 1.24i)21-s − 0.728i·23-s + 0.930·25-s + (0.995 + 0.0912i)27-s − 0.432i·29-s + 0.655·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 + 0.526i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.850 + 0.526i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.37512 - 0.391006i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.37512 - 0.391006i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.57 - 2.55i)T \) |
good | 5 | \( 1 + 1.31iT - 25T^{2} \) |
| 7 | \( 1 - 10.2T + 49T^{2} \) |
| 11 | \( 1 + 16.6iT - 121T^{2} \) |
| 13 | \( 1 + 18.7T + 169T^{2} \) |
| 17 | \( 1 + 4.38iT - 289T^{2} \) |
| 19 | \( 1 - 11.5T + 361T^{2} \) |
| 23 | \( 1 + 16.7iT - 529T^{2} \) |
| 29 | \( 1 + 12.5iT - 841T^{2} \) |
| 31 | \( 1 - 20.3T + 961T^{2} \) |
| 37 | \( 1 - 18.5T + 1.36e3T^{2} \) |
| 41 | \( 1 + 78.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 36.4T + 1.84e3T^{2} \) |
| 47 | \( 1 - 19.9iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 81.3iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 29.9iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 72.0T + 3.72e3T^{2} \) |
| 67 | \( 1 - 56.3T + 4.48e3T^{2} \) |
| 71 | \( 1 - 136. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 80.8T + 5.32e3T^{2} \) |
| 79 | \( 1 - 86.0T + 6.24e3T^{2} \) |
| 83 | \( 1 + 80.4iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 131. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 20.4T + 9.40e3T^{2} \) |
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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
−11.05615057180581107107739091773, −10.32341716935101921979650652146, −9.170484585588316093123314985503, −8.435559483765973055489411724570, −7.37384227672444910821187335093, −5.90818162112486187848081304490, −5.07275884702140877426744349021, −4.33493821574499734565531826597, −2.78771713069567420328667420746, −0.74184620650847777060335736870,
1.43724910744318389921781160570, 2.49307680086248986191771769704, 4.69079525820332987185123725016, 5.15899061921002890053904981823, 6.65903954191368872060343422149, 7.51244521001676639611687291179, 7.990918731506691587010193253484, 9.465680447166868395733633435044, 10.45201341521455311440678916703, 11.39545867342572475843040954273