| L(s) = 1 | − 8·3-s + 5·5-s + 7·7-s + 37·9-s + 68·11-s − 34·13-s − 40·15-s + 74·17-s − 128·19-s − 56·21-s + 80·23-s + 25·25-s − 80·27-s − 286·29-s + 24·31-s − 544·33-s + 35·35-s − 294·37-s + 272·39-s + 66·41-s − 124·43-s + 185·45-s − 312·47-s + 49·49-s − 592·51-s + 34·53-s + 340·55-s + ⋯ |
| L(s) = 1 | − 1.53·3-s + 0.447·5-s + 0.377·7-s + 1.37·9-s + 1.86·11-s − 0.725·13-s − 0.688·15-s + 1.05·17-s − 1.54·19-s − 0.581·21-s + 0.725·23-s + 1/5·25-s − 0.570·27-s − 1.83·29-s + 0.139·31-s − 2.86·33-s + 0.169·35-s − 1.30·37-s + 1.11·39-s + 0.251·41-s − 0.439·43-s + 0.612·45-s − 0.968·47-s + 1/7·49-s − 1.62·51-s + 0.0881·53-s + 0.833·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - p T \) |
| 7 | \( 1 - p T \) |
| good | 3 | \( 1 + 8 T + p^{3} T^{2} \) |
| 11 | \( 1 - 68 T + p^{3} T^{2} \) |
| 13 | \( 1 + 34 T + p^{3} T^{2} \) |
| 17 | \( 1 - 74 T + p^{3} T^{2} \) |
| 19 | \( 1 + 128 T + p^{3} T^{2} \) |
| 23 | \( 1 - 80 T + p^{3} T^{2} \) |
| 29 | \( 1 + 286 T + p^{3} T^{2} \) |
| 31 | \( 1 - 24 T + p^{3} T^{2} \) |
| 37 | \( 1 + 294 T + p^{3} T^{2} \) |
| 41 | \( 1 - 66 T + p^{3} T^{2} \) |
| 43 | \( 1 + 124 T + p^{3} T^{2} \) |
| 47 | \( 1 + 312 T + p^{3} T^{2} \) |
| 53 | \( 1 - 34 T + p^{3} T^{2} \) |
| 59 | \( 1 - 168 T + p^{3} T^{2} \) |
| 61 | \( 1 + 170 T + p^{3} T^{2} \) |
| 67 | \( 1 - 564 T + p^{3} T^{2} \) |
| 71 | \( 1 + 616 T + p^{3} T^{2} \) |
| 73 | \( 1 - 250 T + p^{3} T^{2} \) |
| 79 | \( 1 - 944 T + p^{3} T^{2} \) |
| 83 | \( 1 - 672 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1430 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1270 T + p^{3} T^{2} \) |
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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
−8.337039264137351485949940279377, −7.12616178911984399976034846800, −6.67600058825507359288349372765, −5.90175705257501004587306345552, −5.24900413730443097920658898367, −4.45890338801503005716292837321, −3.57089949391027404796266956684, −1.93320583777730813137430441128, −1.16127608750132625896042765810, 0,
1.16127608750132625896042765810, 1.93320583777730813137430441128, 3.57089949391027404796266956684, 4.45890338801503005716292837321, 5.24900413730443097920658898367, 5.90175705257501004587306345552, 6.67600058825507359288349372765, 7.12616178911984399976034846800, 8.337039264137351485949940279377
