L(s) = 1 | + 0.512i·5-s + 7-s + 1.82i·11-s + 1.80i·13-s − 8.11·17-s − 3.43i·19-s − 3.65·23-s + 4.73·25-s − 7.98i·29-s + 1.56·31-s + 0.512i·35-s − 8.44i·37-s + 2.30·41-s + 10.7i·43-s + 11.3·47-s + ⋯ |
L(s) = 1 | + 0.229i·5-s + 0.377·7-s + 0.551i·11-s + 0.500i·13-s − 1.96·17-s − 0.789i·19-s − 0.762·23-s + 0.947·25-s − 1.48i·29-s + 0.281·31-s + 0.0866i·35-s − 1.38i·37-s + 0.360·41-s + 1.64i·43-s + 1.66·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.972 + 0.232i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.972 + 0.232i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.752308248\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.752308248\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
good | 5 | \( 1 - 0.512iT - 5T^{2} \) |
| 11 | \( 1 - 1.82iT - 11T^{2} \) |
| 13 | \( 1 - 1.80iT - 13T^{2} \) |
| 17 | \( 1 + 8.11T + 17T^{2} \) |
| 19 | \( 1 + 3.43iT - 19T^{2} \) |
| 23 | \( 1 + 3.65T + 23T^{2} \) |
| 29 | \( 1 + 7.98iT - 29T^{2} \) |
| 31 | \( 1 - 1.56T + 31T^{2} \) |
| 37 | \( 1 + 8.44iT - 37T^{2} \) |
| 41 | \( 1 - 2.30T + 41T^{2} \) |
| 43 | \( 1 - 10.7iT - 43T^{2} \) |
| 47 | \( 1 - 11.3T + 47T^{2} \) |
| 53 | \( 1 + 8.87iT - 53T^{2} \) |
| 59 | \( 1 - 2.50iT - 59T^{2} \) |
| 61 | \( 1 + 5.31iT - 61T^{2} \) |
| 67 | \( 1 + 6.44iT - 67T^{2} \) |
| 71 | \( 1 - 9.10T + 71T^{2} \) |
| 73 | \( 1 - 9.24T + 73T^{2} \) |
| 79 | \( 1 + 3.64T + 79T^{2} \) |
| 83 | \( 1 - 4.53iT - 83T^{2} \) |
| 89 | \( 1 - 11.7T + 89T^{2} \) |
| 97 | \( 1 - 15.2T + 97T^{2} \) |
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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
−7.978126829766502062265186330321, −7.33557163459239884368224884538, −6.57736467235413818588822916796, −6.14349433718569690636006437254, −4.94800293035145080642788265130, −4.48418161969256402727171658533, −3.77270729705841322004835258428, −2.39471647809894149454908643624, −2.14203791786127777985349420726, −0.59920809127733180469888515757,
0.78295354212760430029894827203, 1.88683310098123621468421065098, 2.77408010821410357107079016993, 3.73477996929707872313764573829, 4.49827789529072480158014790934, 5.20618629316684384574847936101, 5.96408113802910659006064404454, 6.69022259296538925358983568528, 7.38253641697328047412565182296, 8.252728836447680882084134205085