| L(s) = 1 | + 3-s − 7-s − 2·9-s − 5·13-s − 6·17-s − 19-s − 21-s − 7·23-s − 5·27-s − 10·29-s − 10·31-s + 8·37-s − 5·39-s + 8·41-s − 8·43-s + 6·47-s + 49-s − 6·51-s + 12·53-s − 57-s + 11·59-s + 14·61-s + 2·63-s − 7·69-s + 4·73-s + 5·79-s + 81-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.377·7-s − 2/3·9-s − 1.38·13-s − 1.45·17-s − 0.229·19-s − 0.218·21-s − 1.45·23-s − 0.962·27-s − 1.85·29-s − 1.79·31-s + 1.31·37-s − 0.800·39-s + 1.24·41-s − 1.21·43-s + 0.875·47-s + 1/7·49-s − 0.840·51-s + 1.64·53-s − 0.132·57-s + 1.43·59-s + 1.79·61-s + 0.251·63-s − 0.842·69-s + 0.468·73-s + 0.562·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 169400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 7 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 11 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 - 17 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−13.33122626078970, −13.11529276693770, −12.61271797653509, −12.05125554892793, −11.50341479012452, −11.17133661906378, −10.66475893958221, −9.965626832443404, −9.532796762910367, −9.256037343101098, −8.682363721187929, −8.239433092351763, −7.597366513537827, −7.282597368587751, −6.740902833754419, −6.086580885692579, −5.524082245909297, −5.249219284564383, −4.290900334805032, −3.938521642726624, −3.499238679671776, −2.510035451393819, −2.283330859747567, −1.938271752134893, −0.5760506363973158, 0,
0.5760506363973158, 1.938271752134893, 2.283330859747567, 2.510035451393819, 3.499238679671776, 3.938521642726624, 4.290900334805032, 5.249219284564383, 5.524082245909297, 6.086580885692579, 6.740902833754419, 7.282597368587751, 7.597366513537827, 8.239433092351763, 8.682363721187929, 9.256037343101098, 9.532796762910367, 9.965626832443404, 10.66475893958221, 11.17133661906378, 11.50341479012452, 12.05125554892793, 12.61271797653509, 13.11529276693770, 13.33122626078970
