| L(s) = 1 | + 5-s + 7-s + 2·13-s − 2·17-s − 8·19-s − 8·23-s + 25-s + 6·29-s + 35-s + 2·37-s + 6·41-s − 8·43-s + 8·47-s + 49-s − 2·53-s + 8·59-s + 2·61-s + 2·65-s − 8·67-s + 10·73-s − 16·79-s + 16·83-s − 2·85-s − 10·89-s + 2·91-s − 8·95-s − 14·97-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.377·7-s + 0.554·13-s − 0.485·17-s − 1.83·19-s − 1.66·23-s + 1/5·25-s + 1.11·29-s + 0.169·35-s + 0.328·37-s + 0.937·41-s − 1.21·43-s + 1.16·47-s + 1/7·49-s − 0.274·53-s + 1.04·59-s + 0.256·61-s + 0.248·65-s − 0.977·67-s + 1.17·73-s − 1.80·79-s + 1.75·83-s − 0.216·85-s − 1.05·89-s + 0.209·91-s − 0.820·95-s − 1.42·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 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 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−15.91356234058504, −15.34864996442962, −14.85709456514589, −14.17638263519404, −13.85482358708542, −13.21134962351665, −12.65959884386941, −12.17262900902953, −11.42196231448244, −10.99857161477267, −10.19866020249681, −10.10538049540235, −9.085750228515928, −8.590783317011674, −8.197797424950524, −7.466360880440836, −6.600619724913675, −6.226110248335341, −5.678898321361997, −4.754767909126434, −4.254431732690618, −3.634540912438433, −2.488190466014157, −2.102851059296370, −1.169208060423958, 0,
1.169208060423958, 2.102851059296370, 2.488190466014157, 3.634540912438433, 4.254431732690618, 4.754767909126434, 5.678898321361997, 6.226110248335341, 6.600619724913675, 7.466360880440836, 8.197797424950524, 8.590783317011674, 9.085750228515928, 10.10538049540235, 10.19866020249681, 10.99857161477267, 11.42196231448244, 12.17262900902953, 12.65959884386941, 13.21134962351665, 13.85482358708542, 14.17638263519404, 14.85709456514589, 15.34864996442962, 15.91356234058504
