L(s) = 1 | + 2.96i·3-s + (2.22 + 0.177i)5-s + (−0.115 + 0.115i)7-s − 5.79·9-s + (−2.95 + 2.95i)11-s − 1.55·13-s + (−0.525 + 6.61i)15-s + (0.299 − 0.299i)17-s + (−2.26 + 2.26i)19-s + (−0.341 − 0.341i)21-s + (4.14 + 4.14i)23-s + (4.93 + 0.790i)25-s − 8.28i·27-s + (0.289 + 0.289i)29-s − 4.18i·31-s + ⋯ |
L(s) = 1 | + 1.71i·3-s + (0.996 + 0.0793i)5-s + (−0.0435 + 0.0435i)7-s − 1.93·9-s + (−0.892 + 0.892i)11-s − 0.432·13-s + (−0.135 + 1.70i)15-s + (0.0726 − 0.0726i)17-s + (−0.519 + 0.519i)19-s + (−0.0744 − 0.0744i)21-s + (0.864 + 0.864i)23-s + (0.987 + 0.158i)25-s − 1.59i·27-s + (0.0537 + 0.0537i)29-s − 0.751i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.852 - 0.522i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.852 - 0.522i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.386864 + 1.37259i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.386864 + 1.37259i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-2.22 - 0.177i)T \) |
good | 3 | \( 1 - 2.96iT - 3T^{2} \) |
| 7 | \( 1 + (0.115 - 0.115i)T - 7iT^{2} \) |
| 11 | \( 1 + (2.95 - 2.95i)T - 11iT^{2} \) |
| 13 | \( 1 + 1.55T + 13T^{2} \) |
| 17 | \( 1 + (-0.299 + 0.299i)T - 17iT^{2} \) |
| 19 | \( 1 + (2.26 - 2.26i)T - 19iT^{2} \) |
| 23 | \( 1 + (-4.14 - 4.14i)T + 23iT^{2} \) |
| 29 | \( 1 + (-0.289 - 0.289i)T + 29iT^{2} \) |
| 31 | \( 1 + 4.18iT - 31T^{2} \) |
| 37 | \( 1 + 1.63T + 37T^{2} \) |
| 41 | \( 1 - 7.61iT - 41T^{2} \) |
| 43 | \( 1 - 6.72T + 43T^{2} \) |
| 47 | \( 1 + (4.38 + 4.38i)T + 47iT^{2} \) |
| 53 | \( 1 + 11.4iT - 53T^{2} \) |
| 59 | \( 1 + (-1.63 - 1.63i)T + 59iT^{2} \) |
| 61 | \( 1 + (-1.23 + 1.23i)T - 61iT^{2} \) |
| 67 | \( 1 + 2.49T + 67T^{2} \) |
| 71 | \( 1 - 8.00T + 71T^{2} \) |
| 73 | \( 1 + (-1.12 + 1.12i)T - 73iT^{2} \) |
| 79 | \( 1 + 3.62T + 79T^{2} \) |
| 83 | \( 1 + 1.62iT - 83T^{2} \) |
| 89 | \( 1 - 15.7T + 89T^{2} \) |
| 97 | \( 1 + (-9.69 + 9.69i)T - 97iT^{2} \) |
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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
−10.63935201292867512142708076531, −9.904805495968235054454259191965, −9.604580762766658485567047858477, −8.663072792296976728760111796019, −7.47791553759515892139311292826, −6.14533841729443917449116002079, −5.20255532962984912417526742684, −4.66166327608639006782492325143, −3.37666752676865279711435769727, −2.26138598187087367362768344333,
0.75370939085634236552201166155, 2.14526164299037979015195225452, 2.92503882896414934774289907753, 5.02434052858071085535213852384, 5.90480915928441851354273243105, 6.65690720470573014663205907423, 7.43620398887230990943916484381, 8.433534972965240261746148565738, 9.050118952436220798321804218040, 10.41457526925671812798686306948