| L(s) = 1 | + 5-s − 3·11-s + 13-s + 17-s − 19-s + 25-s + 5·29-s + 4·31-s + 2·37-s − 2·41-s − 10·43-s + 7·47-s + 3·53-s − 3·55-s − 5·59-s + 61-s + 65-s + 9·67-s − 11·71-s + 4·73-s − 10·79-s − 8·83-s + 85-s − 6·89-s − 95-s − 16·97-s + 101-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.904·11-s + 0.277·13-s + 0.242·17-s − 0.229·19-s + 1/5·25-s + 0.928·29-s + 0.718·31-s + 0.328·37-s − 0.312·41-s − 1.52·43-s + 1.02·47-s + 0.412·53-s − 0.404·55-s − 0.650·59-s + 0.128·61-s + 0.124·65-s + 1.09·67-s − 1.30·71-s + 0.468·73-s − 1.12·79-s − 0.878·83-s + 0.108·85-s − 0.635·89-s − 0.102·95-s − 1.62·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 229320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 229320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 11 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 - 9 T + p T^{2} \) |
| 71 | \( 1 + 11 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−13.24420062023707, −12.73891539594319, −12.25625896113074, −11.83578892218508, −11.22967067650501, −10.85584707089113, −10.22417033856780, −10.01161356064214, −9.598591461464553, −8.765609199310615, −8.506704893882027, −8.113002986333069, −7.394583276543907, −7.065376659390693, −6.375795596663982, −6.019294827315290, −5.454631737512648, −4.963874089841386, −4.487598200909811, −3.883734262949001, −3.131479408242789, −2.767374239692955, −2.159046838328847, −1.483018608322851, −0.8306938542767054, 0,
0.8306938542767054, 1.483018608322851, 2.159046838328847, 2.767374239692955, 3.131479408242789, 3.883734262949001, 4.487598200909811, 4.963874089841386, 5.454631737512648, 6.019294827315290, 6.375795596663982, 7.065376659390693, 7.394583276543907, 8.113002986333069, 8.506704893882027, 8.765609199310615, 9.598591461464553, 10.01161356064214, 10.22417033856780, 10.85584707089113, 11.22967067650501, 11.83578892218508, 12.25625896113074, 12.73891539594319, 13.24420062023707
