| L(s) = 1 | + 5-s − 2·11-s − 2·13-s + 4·19-s − 2·23-s + 25-s + 6·29-s − 2·37-s + 12·41-s + 43-s − 2·47-s − 7·49-s − 8·53-s − 2·55-s − 2·59-s + 2·61-s − 2·65-s − 4·67-s − 8·71-s − 14·73-s + 8·79-s + 14·83-s + 6·89-s + 4·95-s + 2·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.603·11-s − 0.554·13-s + 0.917·19-s − 0.417·23-s + 1/5·25-s + 1.11·29-s − 0.328·37-s + 1.87·41-s + 0.152·43-s − 0.291·47-s − 49-s − 1.09·53-s − 0.269·55-s − 0.260·59-s + 0.256·61-s − 0.248·65-s − 0.488·67-s − 0.949·71-s − 1.63·73-s + 0.900·79-s + 1.53·83-s + 0.635·89-s + 0.410·95-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 123840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 123840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 43 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−13.66760763430219, −13.40589790254311, −12.80232204940853, −12.30980401967987, −11.95466161976601, −11.37774725418643, −10.70397181340310, −10.49960588775260, −9.766018873330948, −9.503843460296219, −9.023379715073131, −8.208625213326564, −7.965213318504170, −7.313478461239775, −6.920688975046663, −6.053339203402379, −5.947741359388801, −5.084153857641256, −4.787829322006962, −4.178708561463267, −3.325150858045796, −2.870395459167346, −2.318708807799488, −1.592343402197235, −0.8826832611575973, 0,
0.8826832611575973, 1.592343402197235, 2.318708807799488, 2.870395459167346, 3.325150858045796, 4.178708561463267, 4.787829322006962, 5.084153857641256, 5.947741359388801, 6.053339203402379, 6.920688975046663, 7.313478461239775, 7.965213318504170, 8.208625213326564, 9.023379715073131, 9.503843460296219, 9.766018873330948, 10.49960588775260, 10.70397181340310, 11.37774725418643, 11.95466161976601, 12.30980401967987, 12.80232204940853, 13.40589790254311, 13.66760763430219
