| L(s) = 1 | + 3-s − 5·7-s + 9-s + 11-s − 13-s − 3·17-s − 6·19-s − 5·21-s − 3·23-s + 27-s + 4·29-s + 33-s + 5·37-s − 39-s + 11·41-s − 6·43-s + 18·49-s − 3·51-s + 9·53-s − 6·57-s + 12·59-s + 5·61-s − 5·63-s − 8·67-s − 3·69-s + 13·71-s + 10·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.88·7-s + 1/3·9-s + 0.301·11-s − 0.277·13-s − 0.727·17-s − 1.37·19-s − 1.09·21-s − 0.625·23-s + 0.192·27-s + 0.742·29-s + 0.174·33-s + 0.821·37-s − 0.160·39-s + 1.71·41-s − 0.914·43-s + 18/7·49-s − 0.420·51-s + 1.23·53-s − 0.794·57-s + 1.56·59-s + 0.640·61-s − 0.629·63-s − 0.977·67-s − 0.361·69-s + 1.54·71-s + 1.17·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + 5 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 - 11 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 13 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 17 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 11 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−15.35984758680882, −14.85744283885805, −14.31468973183478, −13.67900201755143, −13.16798439515475, −12.80731238581149, −12.40306642573719, −11.72063323341610, −11.02008682467463, −10.32019405788015, −9.936962541627726, −9.477089257467323, −8.835559401360135, −8.490253242685211, −7.695780232419356, −6.953927363523733, −6.567838155546896, −6.135339964869706, −5.399937404351906, −4.281757193785834, −4.069316156708908, −3.315357511437940, −2.514847117271842, −2.242013870694689, −0.8900634496514382, 0,
0.8900634496514382, 2.242013870694689, 2.514847117271842, 3.315357511437940, 4.069316156708908, 4.281757193785834, 5.399937404351906, 6.135339964869706, 6.567838155546896, 6.953927363523733, 7.695780232419356, 8.490253242685211, 8.835559401360135, 9.477089257467323, 9.936962541627726, 10.32019405788015, 11.02008682467463, 11.72063323341610, 12.40306642573719, 12.80731238581149, 13.16798439515475, 13.67900201755143, 14.31468973183478, 14.85744283885805, 15.35984758680882
