| L(s) = 1 | + 2·5-s − 4·7-s − 2·11-s − 2·13-s + 2·17-s − 2·19-s − 4·23-s − 25-s − 6·29-s − 8·35-s − 10·37-s + 6·41-s − 6·43-s + 8·47-s + 9·49-s − 6·53-s − 4·55-s + 14·59-s − 2·61-s − 4·65-s − 10·67-s − 12·71-s + 14·73-s + 8·77-s − 8·79-s − 6·83-s + 4·85-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 1.51·7-s − 0.603·11-s − 0.554·13-s + 0.485·17-s − 0.458·19-s − 0.834·23-s − 1/5·25-s − 1.11·29-s − 1.35·35-s − 1.64·37-s + 0.937·41-s − 0.914·43-s + 1.16·47-s + 9/7·49-s − 0.824·53-s − 0.539·55-s + 1.82·59-s − 0.256·61-s − 0.496·65-s − 1.22·67-s − 1.42·71-s + 1.63·73-s + 0.911·77-s − 0.900·79-s − 0.658·83-s + 0.433·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 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
−9.604101000606496287963353743481, −8.759776534720553396196404795081, −7.62957697295837367057177448880, −6.80509880453900742773724680138, −5.94298026933117547539540451748, −5.37965797861855942151102168788, −3.98566652809305192679300882554, −2.97266024838274409285785943283, −1.98277685475006180754048515240, 0,
1.98277685475006180754048515240, 2.97266024838274409285785943283, 3.98566652809305192679300882554, 5.37965797861855942151102168788, 5.94298026933117547539540451748, 6.80509880453900742773724680138, 7.62957697295837367057177448880, 8.759776534720553396196404795081, 9.604101000606496287963353743481
