| L(s) = 1 | − 2-s − 7·4-s + 6·7-s + 15·8-s + 43·11-s − 28·13-s − 6·14-s + 41·16-s − 91·17-s − 35·19-s − 43·22-s − 162·23-s + 28·26-s − 42·28-s − 160·29-s + 42·31-s − 161·32-s + 91·34-s − 314·37-s + 35·38-s + 203·41-s + 92·43-s − 301·44-s + 162·46-s − 196·47-s − 307·49-s + 196·52-s + ⋯ |
| L(s) = 1 | − 0.353·2-s − 7/8·4-s + 0.323·7-s + 0.662·8-s + 1.17·11-s − 0.597·13-s − 0.114·14-s + 0.640·16-s − 1.29·17-s − 0.422·19-s − 0.416·22-s − 1.46·23-s + 0.211·26-s − 0.283·28-s − 1.02·29-s + 0.243·31-s − 0.889·32-s + 0.459·34-s − 1.39·37-s + 0.149·38-s + 0.773·41-s + 0.326·43-s − 1.03·44-s + 0.519·46-s − 0.608·47-s − 0.895·49-s + 0.522·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + T + p^{3} T^{2} \) |
| 7 | \( 1 - 6 T + p^{3} T^{2} \) |
| 11 | \( 1 - 43 T + p^{3} T^{2} \) |
| 13 | \( 1 + 28 T + p^{3} T^{2} \) |
| 17 | \( 1 + 91 T + p^{3} T^{2} \) |
| 19 | \( 1 + 35 T + p^{3} T^{2} \) |
| 23 | \( 1 + 162 T + p^{3} T^{2} \) |
| 29 | \( 1 + 160 T + p^{3} T^{2} \) |
| 31 | \( 1 - 42 T + p^{3} T^{2} \) |
| 37 | \( 1 + 314 T + p^{3} T^{2} \) |
| 41 | \( 1 - 203 T + p^{3} T^{2} \) |
| 43 | \( 1 - 92 T + p^{3} T^{2} \) |
| 47 | \( 1 + 196 T + p^{3} T^{2} \) |
| 53 | \( 1 + 82 T + p^{3} T^{2} \) |
| 59 | \( 1 - 280 T + p^{3} T^{2} \) |
| 61 | \( 1 + 518 T + p^{3} T^{2} \) |
| 67 | \( 1 - 141 T + p^{3} T^{2} \) |
| 71 | \( 1 + 412 T + p^{3} T^{2} \) |
| 73 | \( 1 + 763 T + p^{3} T^{2} \) |
| 79 | \( 1 - 510 T + p^{3} T^{2} \) |
| 83 | \( 1 + 777 T + p^{3} T^{2} \) |
| 89 | \( 1 - 945 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1246 T + p^{3} T^{2} \) |
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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
−11.27856174220349058157076922998, −10.17213232277731410724179854550, −9.254912150699130830702310221281, −8.543292530951009284779640037331, −7.43336354897359897108373167150, −6.17641697460274263208818221288, −4.75611059288169044369471395799, −3.88057586572518254425850795176, −1.80151696573087936278175360327, 0,
1.80151696573087936278175360327, 3.88057586572518254425850795176, 4.75611059288169044369471395799, 6.17641697460274263208818221288, 7.43336354897359897108373167150, 8.543292530951009284779640037331, 9.254912150699130830702310221281, 10.17213232277731410724179854550, 11.27856174220349058157076922998
