L(s) = 1 | + (0.224 − 0.224i)2-s + 3.89i·4-s + (−3.44 + 3.44i)7-s + (1.77 + 1.77i)8-s − 11.3·11-s + (5.55 + 5.55i)13-s + 1.55i·14-s − 14.7·16-s + (−17.3 + 17.3i)17-s − 8.69i·19-s + (−2.55 + 2.55i)22-s + (11.5 + 11.5i)23-s + 2.49·26-s + (−13.4 − 13.4i)28-s + 35.1i·29-s + ⋯ |
L(s) = 1 | + (0.112 − 0.112i)2-s + 0.974i·4-s + (−0.492 + 0.492i)7-s + (0.221 + 0.221i)8-s − 1.03·11-s + (0.426 + 0.426i)13-s + 0.110i·14-s − 0.924·16-s + (−1.02 + 1.02i)17-s − 0.457i·19-s + (−0.115 + 0.115i)22-s + (0.502 + 0.502i)23-s + 0.0959·26-s + (−0.480 − 0.480i)28-s + 1.21i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.525 - 0.850i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.535386 + 0.960272i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.535386 + 0.960272i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + (-0.224 + 0.224i)T - 4iT^{2} \) |
| 7 | \( 1 + (3.44 - 3.44i)T - 49iT^{2} \) |
| 11 | \( 1 + 11.3T + 121T^{2} \) |
| 13 | \( 1 + (-5.55 - 5.55i)T + 169iT^{2} \) |
| 17 | \( 1 + (17.3 - 17.3i)T - 289iT^{2} \) |
| 19 | \( 1 + 8.69iT - 361T^{2} \) |
| 23 | \( 1 + (-11.5 - 11.5i)T + 529iT^{2} \) |
| 29 | \( 1 - 35.1iT - 841T^{2} \) |
| 31 | \( 1 - 10.6T + 961T^{2} \) |
| 37 | \( 1 + (-6.04 + 6.04i)T - 1.36e3iT^{2} \) |
| 41 | \( 1 + 0.696T + 1.68e3T^{2} \) |
| 43 | \( 1 + (-26.4 - 26.4i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (-44.2 + 44.2i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + (0.696 + 0.696i)T + 2.80e3iT^{2} \) |
| 59 | \( 1 + 39.9iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 5.90T + 3.72e3T^{2} \) |
| 67 | \( 1 + (-45.1 + 45.1i)T - 4.48e3iT^{2} \) |
| 71 | \( 1 - 68T + 5.04e3T^{2} \) |
| 73 | \( 1 + (77.7 + 77.7i)T + 5.32e3iT^{2} \) |
| 79 | \( 1 - 24.4iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (-13.1 - 13.1i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 - 82.1iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-24.5 + 24.5i)T - 9.40e3iT^{2} \) |
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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
−12.58394605085179503213649033274, −11.39476394926789808720723258612, −10.63673146277486606482652384171, −9.184783549582935469198659225124, −8.486885153823864915675400631952, −7.37966063108100749050732323340, −6.29239232322289397479535755583, −4.86716117192925319806462745489, −3.55654715797918749650014039214, −2.36140027017480890621960571235,
0.55708006345420739640671132517, 2.56426142020865086970385653538, 4.32402725310536356477860026764, 5.46132822901386991078901940701, 6.48374007198794133654159174354, 7.52257805448851540024171845167, 8.857511799366143256330519183974, 9.950508808261636602058759166009, 10.58517369895223074999926546395, 11.49679743513310089915117842575