L(s) = 1 | − 2.53·2-s + 4.41·4-s + 0.879·5-s + 7-s − 6.10·8-s − 2.22·10-s − 3.87·11-s − 5.45·13-s − 2.53·14-s + 6.63·16-s − 1.65·17-s + 2.41·19-s + 3.87·20-s + 9.82·22-s − 3.16·23-s − 4.22·25-s + 13.8·26-s + 4.41·28-s + 6.04·29-s − 4.55·31-s − 4.59·32-s + 4.18·34-s + 0.879·35-s − 4.55·37-s − 6.10·38-s − 5.36·40-s + 1.18·41-s + ⋯ |
L(s) = 1 | − 1.79·2-s + 2.20·4-s + 0.393·5-s + 0.377·7-s − 2.15·8-s − 0.704·10-s − 1.16·11-s − 1.51·13-s − 0.676·14-s + 1.65·16-s − 0.400·17-s + 0.553·19-s + 0.867·20-s + 2.09·22-s − 0.659·23-s − 0.845·25-s + 2.70·26-s + 0.833·28-s + 1.12·29-s − 0.817·31-s − 0.812·32-s + 0.717·34-s + 0.148·35-s − 0.748·37-s − 0.990·38-s − 0.849·40-s + 0.185·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 2 | \( 1 + 2.53T + 2T^{2} \) |
| 5 | \( 1 - 0.879T + 5T^{2} \) |
| 11 | \( 1 + 3.87T + 11T^{2} \) |
| 13 | \( 1 + 5.45T + 13T^{2} \) |
| 17 | \( 1 + 1.65T + 17T^{2} \) |
| 19 | \( 1 - 2.41T + 19T^{2} \) |
| 23 | \( 1 + 3.16T + 23T^{2} \) |
| 29 | \( 1 - 6.04T + 29T^{2} \) |
| 31 | \( 1 + 4.55T + 31T^{2} \) |
| 37 | \( 1 + 4.55T + 37T^{2} \) |
| 41 | \( 1 - 1.18T + 41T^{2} \) |
| 43 | \( 1 - 0.184T + 43T^{2} \) |
| 47 | \( 1 - 1.02T + 47T^{2} \) |
| 53 | \( 1 + 7.29T + 53T^{2} \) |
| 59 | \( 1 + 6.66T + 59T^{2} \) |
| 61 | \( 1 + 2.59T + 61T^{2} \) |
| 67 | \( 1 + 2.95T + 67T^{2} \) |
| 71 | \( 1 - 3.68T + 71T^{2} \) |
| 73 | \( 1 + 12.7T + 73T^{2} \) |
| 79 | \( 1 + 5.95T + 79T^{2} \) |
| 83 | \( 1 - 0.218T + 83T^{2} \) |
| 89 | \( 1 + 11.0T + 89T^{2} \) |
| 97 | \( 1 - 12.5T + 97T^{2} \) |
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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
−10.10230894140699732046751167454, −9.554802214366261132376443105118, −8.577801434617660734396194093914, −7.71982549934738127554574527712, −7.22294042911838076464286174298, −5.97690938482399114685884158899, −4.84680165485764198067841520818, −2.76887134811980147821937033308, −1.85017468245371177253221244307, 0,
1.85017468245371177253221244307, 2.76887134811980147821937033308, 4.84680165485764198067841520818, 5.97690938482399114685884158899, 7.22294042911838076464286174298, 7.71982549934738127554574527712, 8.577801434617660734396194093914, 9.554802214366261132376443105118, 10.10230894140699732046751167454