| L(s) = 1 | − 3-s + 4·5-s + 2·7-s + 9-s + 4·13-s − 4·15-s + 2·17-s − 2·21-s − 6·23-s + 11·25-s − 27-s + 10·29-s − 8·31-s + 8·35-s + 2·37-s − 4·39-s − 2·41-s + 4·43-s + 4·45-s − 2·47-s − 3·49-s − 2·51-s − 4·53-s − 8·61-s + 2·63-s + 16·65-s + 12·67-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.78·5-s + 0.755·7-s + 1/3·9-s + 1.10·13-s − 1.03·15-s + 0.485·17-s − 0.436·21-s − 1.25·23-s + 11/5·25-s − 0.192·27-s + 1.85·29-s − 1.43·31-s + 1.35·35-s + 0.328·37-s − 0.640·39-s − 0.312·41-s + 0.609·43-s + 0.596·45-s − 0.291·47-s − 3/7·49-s − 0.280·51-s − 0.549·53-s − 1.02·61-s + 0.251·63-s + 1.98·65-s + 1.46·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.608429137\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.608429137\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 - 4 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 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
−15.58788075743856, −14.69411249013185, −14.30441667315541, −13.80975017438886, −13.48524716241924, −12.67211034220697, −12.38674508069479, −11.55070149405050, −10.99620029308159, −10.56344520030898, −9.972104898309076, −9.575465915313594, −8.830466093894065, −8.312651465688484, −7.643241075820770, −6.759526326439095, −6.189323614249357, −5.898935350736724, −5.209881997430408, −4.733064897051347, −3.860250286552194, −2.981546086652241, −2.038998607525309, −1.591190554332541, −0.8354460524747393,
0.8354460524747393, 1.591190554332541, 2.038998607525309, 2.981546086652241, 3.860250286552194, 4.733064897051347, 5.209881997430408, 5.898935350736724, 6.189323614249357, 6.759526326439095, 7.643241075820770, 8.312651465688484, 8.830466093894065, 9.575465915313594, 9.972104898309076, 10.56344520030898, 10.99620029308159, 11.55070149405050, 12.38674508069479, 12.67211034220697, 13.48524716241924, 13.80975017438886, 14.30441667315541, 14.69411249013185, 15.58788075743856
