| L(s) = 1 | − 2·3-s − 2·7-s + 9-s + 4·13-s − 4·19-s + 4·21-s − 2·23-s + 4·27-s + 2·29-s − 4·37-s − 8·39-s − 2·41-s − 6·43-s − 6·47-s − 3·49-s + 4·53-s + 8·57-s + 12·59-s − 10·61-s − 2·63-s − 14·67-s + 4·69-s + 8·71-s − 8·73-s − 16·79-s − 11·81-s + 2·83-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.755·7-s + 1/3·9-s + 1.10·13-s − 0.917·19-s + 0.872·21-s − 0.417·23-s + 0.769·27-s + 0.371·29-s − 0.657·37-s − 1.28·39-s − 0.312·41-s − 0.914·43-s − 0.875·47-s − 3/7·49-s + 0.549·53-s + 1.05·57-s + 1.56·59-s − 1.28·61-s − 0.251·63-s − 1.71·67-s + 0.481·69-s + 0.949·71-s − 0.936·73-s − 1.80·79-s − 1.22·81-s + 0.219·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 193600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 193600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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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
−13.24024361665791, −12.91014251771401, −12.26562277141636, −11.87562775359991, −11.52655071778470, −10.96304375048957, −10.57109891880623, −10.12625751983999, −9.786396327106176, −8.921783523201719, −8.637753092904532, −8.223753821129243, −7.422860658465480, −6.898979232348464, −6.393687088387417, −6.107540098058854, −5.731189893680069, −5.009554523361705, −4.603401456437356, −3.943030241920283, −3.366106700365926, −2.909901694906979, −1.998154241378175, −1.423567196333817, −0.5985251184035709, 0,
0.5985251184035709, 1.423567196333817, 1.998154241378175, 2.909901694906979, 3.366106700365926, 3.943030241920283, 4.603401456437356, 5.009554523361705, 5.731189893680069, 6.107540098058854, 6.393687088387417, 6.898979232348464, 7.422860658465480, 8.223753821129243, 8.637753092904532, 8.921783523201719, 9.786396327106176, 10.12625751983999, 10.57109891880623, 10.96304375048957, 11.52655071778470, 11.87562775359991, 12.26562277141636, 12.91014251771401, 13.24024361665791
