| L(s) = 1 | − 2·3-s − 4·7-s + 9-s − 2·11-s + 2·13-s + 2·17-s + 2·19-s + 8·21-s + 4·23-s + 4·27-s + 6·29-s + 4·33-s + 10·37-s − 4·39-s − 6·41-s − 6·43-s − 8·47-s + 9·49-s − 4·51-s − 6·53-s − 4·57-s + 14·59-s − 2·61-s − 4·63-s − 10·67-s − 8·69-s − 12·71-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 1.51·7-s + 1/3·9-s − 0.603·11-s + 0.554·13-s + 0.485·17-s + 0.458·19-s + 1.74·21-s + 0.834·23-s + 0.769·27-s + 1.11·29-s + 0.696·33-s + 1.64·37-s − 0.640·39-s − 0.937·41-s − 0.914·43-s − 1.16·47-s + 9/7·49-s − 0.560·51-s − 0.824·53-s − 0.529·57-s + 1.82·59-s − 0.256·61-s − 0.503·63-s − 1.22·67-s − 0.963·69-s − 1.42·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−8.314943684266841113419279617744, −7.34124243383498037404661094562, −6.51923649393196519370645023678, −6.14145104329035945082286838607, −5.35153294565613657214837064488, −4.61358541606893960058803225823, −3.37287692086053595250123961203, −2.81988983474319788780220066864, −1.09196022553267944315738745219, 0,
1.09196022553267944315738745219, 2.81988983474319788780220066864, 3.37287692086053595250123961203, 4.61358541606893960058803225823, 5.35153294565613657214837064488, 6.14145104329035945082286838607, 6.51923649393196519370645023678, 7.34124243383498037404661094562, 8.314943684266841113419279617744
