| L(s) = 1 | + 0.436·5-s − i·7-s + 2.73i·11-s − 1.64i·13-s + 5.22i·17-s + 5.99·19-s + 7.68·23-s − 4.80·25-s − 4.94·29-s − 9.77i·31-s − 0.436i·35-s + 3.81i·37-s − 9.74i·41-s + 8.84·43-s − 4.54·47-s + ⋯ |
| L(s) = 1 | + 0.195·5-s − 0.377i·7-s + 0.825i·11-s − 0.456i·13-s + 1.26i·17-s + 1.37·19-s + 1.60·23-s − 0.961·25-s − 0.918·29-s − 1.75i·31-s − 0.0737i·35-s + 0.627i·37-s − 1.52i·41-s + 1.34·43-s − 0.663·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.940 - 0.340i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.940 - 0.340i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.107170850\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.107170850\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + iT \) |
| good | 5 | \( 1 - 0.436T + 5T^{2} \) |
| 11 | \( 1 - 2.73iT - 11T^{2} \) |
| 13 | \( 1 + 1.64iT - 13T^{2} \) |
| 17 | \( 1 - 5.22iT - 17T^{2} \) |
| 19 | \( 1 - 5.99T + 19T^{2} \) |
| 23 | \( 1 - 7.68T + 23T^{2} \) |
| 29 | \( 1 + 4.94T + 29T^{2} \) |
| 31 | \( 1 + 9.77iT - 31T^{2} \) |
| 37 | \( 1 - 3.81iT - 37T^{2} \) |
| 41 | \( 1 + 9.74iT - 41T^{2} \) |
| 43 | \( 1 - 8.84T + 43T^{2} \) |
| 47 | \( 1 + 4.54T + 47T^{2} \) |
| 53 | \( 1 - 9.94T + 53T^{2} \) |
| 59 | \( 1 - 13.2iT - 59T^{2} \) |
| 61 | \( 1 - 6.26iT - 61T^{2} \) |
| 67 | \( 1 + 9.42T + 67T^{2} \) |
| 71 | \( 1 - 9.76T + 71T^{2} \) |
| 73 | \( 1 + 9.14T + 73T^{2} \) |
| 79 | \( 1 - 5.19iT - 79T^{2} \) |
| 83 | \( 1 + 0.227iT - 83T^{2} \) |
| 89 | \( 1 + 2.46iT - 89T^{2} \) |
| 97 | \( 1 + 3.37T + 97T^{2} \) |
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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
−7.87051434836180027096171950729, −7.48673597757959707070401869004, −6.84146348638702351389770368544, −5.75572603230083455431524838723, −5.47826602539408827470969620665, −4.34276888530636408271586430986, −3.80565044530676261912975792430, −2.80822952049659278029679260557, −1.87041144849331790268025760099, −0.870247692780859077417944506116,
0.70408771009388275660510121273, 1.74780007529477644136704064834, 2.95083586612624418448994668246, 3.31027696185316886867196140215, 4.52899264278391338747421663697, 5.28905724314204069723139212266, 5.72883402972383824679458685309, 6.73907136937075130470651786970, 7.25421830752026719779043813325, 8.010696382714398018788928964733
