L(s) = 1 | − 1.34·5-s − 7-s + 1.65·11-s − 3.36·13-s − 0.467·17-s + 3.22·19-s + 8.94·23-s − 3.18·25-s − 6.26·29-s − 9.23·31-s + 1.34·35-s + 9.23·37-s − 3.41·41-s + 4.41·43-s + 9.35·47-s + 49-s + 0.573·53-s − 2.22·55-s − 10.3·59-s + 7.63·61-s + 4.53·65-s − 0.596·67-s − 0.554·71-s + 2.04·73-s − 1.65·77-s + 2.40·79-s − 15.0·83-s + ⋯ |
L(s) = 1 | − 0.602·5-s − 0.377·7-s + 0.498·11-s − 0.934·13-s − 0.113·17-s + 0.740·19-s + 1.86·23-s − 0.636·25-s − 1.16·29-s − 1.65·31-s + 0.227·35-s + 1.51·37-s − 0.532·41-s + 0.672·43-s + 1.36·47-s + 0.142·49-s + 0.0788·53-s − 0.300·55-s − 1.35·59-s + 0.977·61-s + 0.563·65-s − 0.0728·67-s − 0.0657·71-s + 0.239·73-s − 0.188·77-s + 0.270·79-s − 1.65·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9072 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 5 | \( 1 + 1.34T + 5T^{2} \) |
| 11 | \( 1 - 1.65T + 11T^{2} \) |
| 13 | \( 1 + 3.36T + 13T^{2} \) |
| 17 | \( 1 + 0.467T + 17T^{2} \) |
| 19 | \( 1 - 3.22T + 19T^{2} \) |
| 23 | \( 1 - 8.94T + 23T^{2} \) |
| 29 | \( 1 + 6.26T + 29T^{2} \) |
| 31 | \( 1 + 9.23T + 31T^{2} \) |
| 37 | \( 1 - 9.23T + 37T^{2} \) |
| 41 | \( 1 + 3.41T + 41T^{2} \) |
| 43 | \( 1 - 4.41T + 43T^{2} \) |
| 47 | \( 1 - 9.35T + 47T^{2} \) |
| 53 | \( 1 - 0.573T + 53T^{2} \) |
| 59 | \( 1 + 10.3T + 59T^{2} \) |
| 61 | \( 1 - 7.63T + 61T^{2} \) |
| 67 | \( 1 + 0.596T + 67T^{2} \) |
| 71 | \( 1 + 0.554T + 71T^{2} \) |
| 73 | \( 1 - 2.04T + 73T^{2} \) |
| 79 | \( 1 - 2.40T + 79T^{2} \) |
| 83 | \( 1 + 15.0T + 83T^{2} \) |
| 89 | \( 1 + 9.08T + 89T^{2} \) |
| 97 | \( 1 + 1.89T + 97T^{2} \) |
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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
−7.25834621573529235719225440153, −7.03489576814038167285532160100, −5.91975648131135809760864526084, −5.37652170633119025062827982768, −4.52463003190870684907048557040, −3.82396398923480200610277172333, −3.12953172606560302081349864140, −2.27864963418210741362490365453, −1.12841708317702186472040900532, 0,
1.12841708317702186472040900532, 2.27864963418210741362490365453, 3.12953172606560302081349864140, 3.82396398923480200610277172333, 4.52463003190870684907048557040, 5.37652170633119025062827982768, 5.91975648131135809760864526084, 7.03489576814038167285532160100, 7.25834621573529235719225440153