| L(s) = 1 | − 2·3-s + 5-s − 7-s + 9-s − 4·11-s + 13-s − 2·15-s − 2·17-s − 19-s + 2·21-s − 7·23-s − 4·25-s + 4·27-s − 5·29-s − 9·31-s + 8·33-s − 35-s − 2·37-s − 2·39-s + 2·41-s + 43-s + 45-s + 9·47-s + 49-s + 4·51-s + 3·53-s − 4·55-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.447·5-s − 0.377·7-s + 1/3·9-s − 1.20·11-s + 0.277·13-s − 0.516·15-s − 0.485·17-s − 0.229·19-s + 0.436·21-s − 1.45·23-s − 4/5·25-s + 0.769·27-s − 0.928·29-s − 1.61·31-s + 1.39·33-s − 0.169·35-s − 0.328·37-s − 0.320·39-s + 0.312·41-s + 0.152·43-s + 0.149·45-s + 1.31·47-s + 1/7·49-s + 0.560·51-s + 0.412·53-s − 0.539·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 364 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 364 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 7 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + 9 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + 14 T + p T^{2} \) |
| 73 | \( 1 - 3 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 - 5 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−10.90103601626300724499148611895, −10.30495888961229291132172615214, −9.300313317494721558051107341205, −8.083973154130150661606579272593, −6.96069631485786572428198608010, −5.82577942199793673176864493442, −5.44176260711429074122925536322, −3.96547517025535220039746450659, −2.23637196597265827007393133486, 0,
2.23637196597265827007393133486, 3.96547517025535220039746450659, 5.44176260711429074122925536322, 5.82577942199793673176864493442, 6.96069631485786572428198608010, 8.083973154130150661606579272593, 9.300313317494721558051107341205, 10.30495888961229291132172615214, 10.90103601626300724499148611895
