| L(s) = 1 | + 3-s + 9-s + 13-s − 2·17-s + 4·19-s + 27-s − 6·29-s − 2·37-s + 39-s + 6·41-s − 12·43-s + 4·47-s − 7·49-s − 2·51-s + 6·53-s + 4·57-s + 8·59-s + 2·61-s + 4·67-s − 12·71-s + 14·73-s + 81-s + 8·83-s − 6·87-s − 18·89-s + 6·97-s + 101-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s + 0.277·13-s − 0.485·17-s + 0.917·19-s + 0.192·27-s − 1.11·29-s − 0.328·37-s + 0.160·39-s + 0.937·41-s − 1.82·43-s + 0.583·47-s − 49-s − 0.280·51-s + 0.824·53-s + 0.529·57-s + 1.04·59-s + 0.256·61-s + 0.488·67-s − 1.42·71-s + 1.63·73-s + 1/9·81-s + 0.878·83-s − 0.643·87-s − 1.90·89-s + 0.609·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 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 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−14.45718118180739, −14.05444608804500, −13.46499553446885, −13.12535064696924, −12.65356749021909, −11.91260054269116, −11.54062986491105, −10.97774060571342, −10.43600607429488, −9.789458588884161, −9.430663149916475, −8.883268360643275, −8.312153240899181, −7.883277520100892, −7.217126332647470, −6.812661799253208, −6.168827635306641, −5.392830755952458, −5.075155863695999, −4.144741516985723, −3.763531897006951, −3.094117120852638, −2.450116849290309, −1.755164770590605, −1.054999794665849, 0,
1.054999794665849, 1.755164770590605, 2.450116849290309, 3.094117120852638, 3.763531897006951, 4.144741516985723, 5.075155863695999, 5.392830755952458, 6.168827635306641, 6.812661799253208, 7.217126332647470, 7.883277520100892, 8.312153240899181, 8.883268360643275, 9.430663149916475, 9.789458588884161, 10.43600607429488, 10.97774060571342, 11.54062986491105, 11.91260054269116, 12.65356749021909, 13.12535064696924, 13.46499553446885, 14.05444608804500, 14.45718118180739
