L(s) = 1 | − 3·5-s + 7-s + 11-s + 2·13-s − 2·17-s − 3·19-s + 23-s + 4·25-s + 2·29-s − 5·31-s − 3·35-s + 7·37-s − 7·41-s + 8·43-s − 4·47-s + 49-s + 4·53-s − 3·55-s − 4·59-s + 4·61-s − 6·65-s + 2·67-s − 3·71-s + 77-s + 4·79-s − 2·83-s + 6·85-s + ⋯ |
L(s) = 1 | − 1.34·5-s + 0.377·7-s + 0.301·11-s + 0.554·13-s − 0.485·17-s − 0.688·19-s + 0.208·23-s + 4/5·25-s + 0.371·29-s − 0.898·31-s − 0.507·35-s + 1.15·37-s − 1.09·41-s + 1.21·43-s − 0.583·47-s + 1/7·49-s + 0.549·53-s − 0.404·55-s − 0.520·59-s + 0.512·61-s − 0.744·65-s + 0.244·67-s − 0.356·71-s + 0.113·77-s + 0.450·79-s − 0.219·83-s + 0.650·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 + 7 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 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
−7.78647946075266095106869546071, −7.09040679489009066280908180885, −6.43023330892861486057843263388, −5.55404355303695336094919181167, −4.58892478666671379745899471265, −4.08828945983619711306818541888, −3.40801457741173171560993204111, −2.36509861745557567121323901143, −1.20070144905169503309757912188, 0,
1.20070144905169503309757912188, 2.36509861745557567121323901143, 3.40801457741173171560993204111, 4.08828945983619711306818541888, 4.58892478666671379745899471265, 5.55404355303695336094919181167, 6.43023330892861486057843263388, 7.09040679489009066280908180885, 7.78647946075266095106869546071