L(s) = 1 | + 2·7-s + 4·11-s + 13-s − 2·19-s + 2·23-s − 4·29-s − 4·31-s + 2·37-s + 6·41-s − 4·43-s − 8·47-s − 3·49-s − 2·53-s + 4·59-s − 2·61-s + 8·67-s + 8·71-s + 4·73-s + 8·77-s − 8·79-s − 12·83-s − 6·89-s + 2·91-s − 4·97-s + 101-s + 103-s + 107-s + ⋯ |
L(s) = 1 | + 0.755·7-s + 1.20·11-s + 0.277·13-s − 0.458·19-s + 0.417·23-s − 0.742·29-s − 0.718·31-s + 0.328·37-s + 0.937·41-s − 0.609·43-s − 1.16·47-s − 3/7·49-s − 0.274·53-s + 0.520·59-s − 0.256·61-s + 0.977·67-s + 0.949·71-s + 0.468·73-s + 0.911·77-s − 0.900·79-s − 1.31·83-s − 0.635·89-s + 0.209·91-s − 0.406·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−14.67391271881716, −14.50743814670608, −13.95266417796353, −13.26446178502334, −12.81845944082502, −12.30430939057194, −11.63161956333586, −11.13401438607651, −11.04008181379718, −10.13489868539618, −9.524529725395410, −9.159719577725772, −8.498179713135247, −8.064067358393256, −7.468258061746150, −6.681823487770853, −6.491652298904671, −5.559046148297341, −5.189683331128381, −4.316337515979324, −3.981496926673180, −3.266702292418285, −2.423864986007324, −1.621330761364646, −1.185397190061214, 0,
1.185397190061214, 1.621330761364646, 2.423864986007324, 3.266702292418285, 3.981496926673180, 4.316337515979324, 5.189683331128381, 5.559046148297341, 6.491652298904671, 6.681823487770853, 7.468258061746150, 8.064067358393256, 8.498179713135247, 9.159719577725772, 9.524529725395410, 10.13489868539618, 11.04008181379718, 11.13401438607651, 11.63161956333586, 12.30430939057194, 12.81845944082502, 13.26446178502334, 13.95266417796353, 14.50743814670608, 14.67391271881716