L(s) = 1 | + (0.516 − 1.31i)2-s + (−1.46 − 1.36i)4-s + (−2.07 − 0.841i)5-s + (−1.13 + 1.13i)7-s + (−2.54 + 1.22i)8-s + (−2.17 + 2.29i)10-s + (2.32 + 2.32i)11-s + 1.36i·13-s + (0.911 + 2.08i)14-s + (0.297 + 3.98i)16-s + (−5.25 + 5.25i)17-s + (−3.69 − 3.69i)19-s + (1.89 + 4.05i)20-s + (4.25 − 1.85i)22-s + (0.911 + 0.911i)23-s + ⋯ |
L(s) = 1 | + (0.365 − 0.930i)2-s + (−0.732 − 0.680i)4-s + (−0.926 − 0.376i)5-s + (−0.430 + 0.430i)7-s + (−0.901 + 0.433i)8-s + (−0.688 + 0.724i)10-s + (0.700 + 0.700i)11-s + 0.378i·13-s + (0.243 + 0.558i)14-s + (0.0744 + 0.997i)16-s + (−1.27 + 1.27i)17-s + (−0.848 − 0.848i)19-s + (0.422 + 0.906i)20-s + (0.907 − 0.395i)22-s + (0.189 + 0.189i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.454 - 0.890i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.454 - 0.890i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.433983 + 0.265619i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.433983 + 0.265619i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.516 + 1.31i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (2.07 + 0.841i)T \) |
good | 7 | \( 1 + (1.13 - 1.13i)T - 7iT^{2} \) |
| 11 | \( 1 + (-2.32 - 2.32i)T + 11iT^{2} \) |
| 13 | \( 1 - 1.36iT - 13T^{2} \) |
| 17 | \( 1 + (5.25 - 5.25i)T - 17iT^{2} \) |
| 19 | \( 1 + (3.69 + 3.69i)T + 19iT^{2} \) |
| 23 | \( 1 + (-0.911 - 0.911i)T + 23iT^{2} \) |
| 29 | \( 1 + (-2.37 + 2.37i)T - 29iT^{2} \) |
| 31 | \( 1 + 0.242iT - 31T^{2} \) |
| 37 | \( 1 - 3.34iT - 37T^{2} \) |
| 41 | \( 1 - 2.66iT - 41T^{2} \) |
| 43 | \( 1 - 9.04iT - 43T^{2} \) |
| 47 | \( 1 + (7.87 + 7.87i)T + 47iT^{2} \) |
| 53 | \( 1 - 5.80T + 53T^{2} \) |
| 59 | \( 1 + (5.91 - 5.91i)T - 59iT^{2} \) |
| 61 | \( 1 + (6.67 + 6.67i)T + 61iT^{2} \) |
| 67 | \( 1 - 4.54iT - 67T^{2} \) |
| 71 | \( 1 + 15.4T + 71T^{2} \) |
| 73 | \( 1 + (1.49 - 1.49i)T - 73iT^{2} \) |
| 79 | \( 1 + 10.3T + 79T^{2} \) |
| 83 | \( 1 + 3.26T + 83T^{2} \) |
| 89 | \( 1 + 9.77T + 89T^{2} \) |
| 97 | \( 1 + (1.63 - 1.63i)T - 97iT^{2} \) |
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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
−10.78017347323333377078941627734, −9.735872869219360795552413828692, −8.913060677305811249268406630251, −8.371462896310872926471288182533, −6.88596332340198038004466779565, −6.08029248057805756833385692900, −4.55082227099745991807945890616, −4.26709790734296737981614829992, −2.96901056837920697978508107191, −1.65934967960976533025809081225,
0.23470889926297489077378308303, 2.98884959021183300916978903378, 3.88855834031085107050435788654, 4.69391745286133140685625429313, 6.01789317718488279383152268657, 6.81621647900449118778113854449, 7.41397234792562738047738921748, 8.482046858627582436480965669565, 9.023680798232490029877963716190, 10.26617049927382666203568330597