L(s) = 1 | + 2·5-s − 7-s + 4·11-s − 2·13-s − 17-s − 4·19-s + 8·23-s − 25-s − 6·29-s − 2·35-s − 2·37-s − 10·41-s + 4·43-s + 49-s − 6·53-s + 8·55-s − 4·59-s + 6·61-s − 4·65-s + 12·67-s − 8·71-s − 6·73-s − 4·77-s − 12·83-s − 2·85-s + 6·89-s + 2·91-s + ⋯ |
L(s) = 1 | + 0.894·5-s − 0.377·7-s + 1.20·11-s − 0.554·13-s − 0.242·17-s − 0.917·19-s + 1.66·23-s − 1/5·25-s − 1.11·29-s − 0.338·35-s − 0.328·37-s − 1.56·41-s + 0.609·43-s + 1/7·49-s − 0.824·53-s + 1.07·55-s − 0.520·59-s + 0.768·61-s − 0.496·65-s + 1.46·67-s − 0.949·71-s − 0.702·73-s − 0.455·77-s − 1.31·83-s − 0.216·85-s + 0.635·89-s + 0.209·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17136 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 17 | \( 1 + T \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−16.20821827652153, −15.54212731498391, −14.87332662929013, −14.58636613975909, −13.97545338584250, −13.19388857177400, −13.07360582415701, −12.29377609907061, −11.72558754204730, −11.07404884725368, −10.54327147585940, −9.817439513398242, −9.374801428387397, −8.950752430526800, −8.310901848676111, −7.352478448384198, −6.791031960483998, −6.396974611411510, −5.639814806410167, −5.056014856315318, −4.254867614712729, −3.571263667526545, −2.753180188848477, −1.952989286079306, −1.283207630973308, 0,
1.283207630973308, 1.952989286079306, 2.753180188848477, 3.571263667526545, 4.254867614712729, 5.056014856315318, 5.639814806410167, 6.396974611411510, 6.791031960483998, 7.352478448384198, 8.310901848676111, 8.950752430526800, 9.374801428387397, 9.817439513398242, 10.54327147585940, 11.07404884725368, 11.72558754204730, 12.29377609907061, 13.07360582415701, 13.19388857177400, 13.97545338584250, 14.58636613975909, 14.87332662929013, 15.54212731498391, 16.20821827652153