| L(s) = 1 | − 3.95i·5-s − 1.63·7-s + 4.82i·11-s − 5.59i·13-s + 0.828·17-s − 2.82i·19-s + 7.91·23-s − 10.6·25-s − 7.23i·29-s − 1.63·31-s + 6.48i·35-s − 2.31i·37-s + 3.17·41-s − 4.48i·43-s − 7.91·47-s + ⋯ |
| L(s) = 1 | − 1.76i·5-s − 0.619·7-s + 1.45i·11-s − 1.55i·13-s + 0.200·17-s − 0.648i·19-s + 1.65·23-s − 2.13·25-s − 1.34i·29-s − 0.294·31-s + 1.09i·35-s − 0.381i·37-s + 0.495·41-s − 0.683i·43-s − 1.15·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.015713098\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.015713098\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 3.95iT - 5T^{2} \) |
| 7 | \( 1 + 1.63T + 7T^{2} \) |
| 11 | \( 1 - 4.82iT - 11T^{2} \) |
| 13 | \( 1 + 5.59iT - 13T^{2} \) |
| 17 | \( 1 - 0.828T + 17T^{2} \) |
| 19 | \( 1 + 2.82iT - 19T^{2} \) |
| 23 | \( 1 - 7.91T + 23T^{2} \) |
| 29 | \( 1 + 7.23iT - 29T^{2} \) |
| 31 | \( 1 + 1.63T + 31T^{2} \) |
| 37 | \( 1 + 2.31iT - 37T^{2} \) |
| 41 | \( 1 - 3.17T + 41T^{2} \) |
| 43 | \( 1 + 4.48iT - 43T^{2} \) |
| 47 | \( 1 + 7.91T + 47T^{2} \) |
| 53 | \( 1 - 0.678iT - 53T^{2} \) |
| 59 | \( 1 - 9.65iT - 59T^{2} \) |
| 61 | \( 1 - 2.31iT - 61T^{2} \) |
| 67 | \( 1 + 13.6iT - 67T^{2} \) |
| 71 | \( 1 + 3.27T + 71T^{2} \) |
| 73 | \( 1 + 4T + 73T^{2} \) |
| 79 | \( 1 - 1.63T + 79T^{2} \) |
| 83 | \( 1 + 8.82iT - 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 - 11.6T + 97T^{2} \) |
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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
−7.86431934148108952846533664192, −7.48557784541778267349554266953, −6.45066798985329048194231406145, −5.54379150029884774883484633069, −4.96402995096335364240950284104, −4.43364332924369855956464684765, −3.39007878332257995113293116285, −2.35862137508168274003751341911, −1.20247683065913050989777548901, −0.29693479142653576316832613149,
1.47045833980917682438978362879, 2.72681006169350691194071019763, 3.26320161692829267630621881360, 3.83871585097276583740262782857, 5.08698067116814768318744444571, 6.05471518253085534605598209862, 6.62071404250718217114145725178, 6.95570610947227663003821003126, 7.84409982601559894254295107612, 8.736995131140879874034451105200
