L(s) = 1 | − 0.206·5-s − 3.03i·7-s − 5.02i·11-s + 4.03i·13-s + 4.81i·17-s − 1.03·19-s − 5.43·23-s − 4.95·25-s − 2.90·29-s + 1.23i·31-s + 0.629i·35-s + 8.77i·37-s − 0.979i·41-s + 3.18·43-s + 3.26·47-s + ⋯ |
L(s) = 1 | − 0.0925·5-s − 1.14i·7-s − 1.51i·11-s + 1.12i·13-s + 1.16i·17-s − 0.238·19-s − 1.13·23-s − 0.991·25-s − 0.539·29-s + 0.222i·31-s + 0.106i·35-s + 1.44i·37-s − 0.152i·41-s + 0.485·43-s + 0.476·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.258 - 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.258 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.019615846\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.019615846\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 0.206T + 5T^{2} \) |
| 7 | \( 1 + 3.03iT - 7T^{2} \) |
| 11 | \( 1 + 5.02iT - 11T^{2} \) |
| 13 | \( 1 - 4.03iT - 13T^{2} \) |
| 17 | \( 1 - 4.81iT - 17T^{2} \) |
| 19 | \( 1 + 1.03T + 19T^{2} \) |
| 23 | \( 1 + 5.43T + 23T^{2} \) |
| 29 | \( 1 + 2.90T + 29T^{2} \) |
| 31 | \( 1 - 1.23iT - 31T^{2} \) |
| 37 | \( 1 - 8.77iT - 37T^{2} \) |
| 41 | \( 1 + 0.979iT - 41T^{2} \) |
| 43 | \( 1 - 3.18T + 43T^{2} \) |
| 47 | \( 1 - 3.26T + 47T^{2} \) |
| 53 | \( 1 - 7.91T + 53T^{2} \) |
| 59 | \( 1 - 14.8iT - 59T^{2} \) |
| 61 | \( 1 - 9.53iT - 61T^{2} \) |
| 67 | \( 1 - 0.735T + 67T^{2} \) |
| 71 | \( 1 + 2.33T + 71T^{2} \) |
| 73 | \( 1 + 10.0T + 73T^{2} \) |
| 79 | \( 1 - 14.1iT - 79T^{2} \) |
| 83 | \( 1 + 4.23iT - 83T^{2} \) |
| 89 | \( 1 + 11.6iT - 89T^{2} \) |
| 97 | \( 1 - 4.84T + 97T^{2} \) |
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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
−8.475485431177941173503253627950, −7.59825024122574061895511796435, −7.00470500867356632560892942746, −6.05142494351870878627083722568, −5.77278627527722959090188811006, −4.28530282446337209919974316321, −4.07719010360158992497821879127, −3.21395360525272090453997362904, −1.97554219773220097136611558736, −1.00553005088229000165310067622,
0.29648270441128732308810374175, 2.00702062701855149398194460148, 2.41086314828069537609587465134, 3.53372483860846214112646200662, 4.41625638034502947686545289667, 5.27591955074685336743104928530, 5.74263956667456223687774180434, 6.60957508761623345328592859566, 7.65381978819145310809741635764, 7.76113482706101432291297790499