L(s) = 1 | − 2·5-s − 7-s − 4·11-s − 6·13-s − 6·17-s + 19-s − 8·23-s − 25-s + 6·29-s + 2·35-s − 2·37-s − 6·41-s − 4·43-s − 4·47-s + 49-s − 2·53-s + 8·55-s − 12·59-s − 10·61-s + 12·65-s + 8·67-s + 10·73-s + 4·77-s − 12·79-s − 16·83-s + 12·85-s − 6·89-s + ⋯ |
L(s) = 1 | − 0.894·5-s − 0.377·7-s − 1.20·11-s − 1.66·13-s − 1.45·17-s + 0.229·19-s − 1.66·23-s − 1/5·25-s + 1.11·29-s + 0.338·35-s − 0.328·37-s − 0.937·41-s − 0.609·43-s − 0.583·47-s + 1/7·49-s − 0.274·53-s + 1.07·55-s − 1.56·59-s − 1.28·61-s + 1.48·65-s + 0.977·67-s + 1.17·73-s + 0.455·77-s − 1.35·79-s − 1.75·83-s + 1.30·85-s − 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 - T \) |
good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 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 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−7.07428010267465136223527821042, −6.46679888409992015778833792006, −5.57594578656304690056296684413, −4.72846187195492181107289242776, −4.39848927071548747070630124612, −3.36135041346081754443511974783, −2.63866021982877608365080940704, −1.91517493932734500492014116626, 0, 0,
1.91517493932734500492014116626, 2.63866021982877608365080940704, 3.36135041346081754443511974783, 4.39848927071548747070630124612, 4.72846187195492181107289242776, 5.57594578656304690056296684413, 6.46679888409992015778833792006, 7.07428010267465136223527821042