| L(s) = 1 | + 3-s − 5-s − 3·7-s + 9-s + 11-s + 13-s − 15-s − 17-s + 5·19-s − 3·21-s − 9·23-s + 25-s + 27-s − 2·29-s − 7·31-s + 33-s + 3·35-s − 3·37-s + 39-s − 2·41-s + 8·43-s − 45-s + 6·47-s + 2·49-s − 51-s + 2·53-s − 55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 1.13·7-s + 1/3·9-s + 0.301·11-s + 0.277·13-s − 0.258·15-s − 0.242·17-s + 1.14·19-s − 0.654·21-s − 1.87·23-s + 1/5·25-s + 0.192·27-s − 0.371·29-s − 1.25·31-s + 0.174·33-s + 0.507·35-s − 0.493·37-s + 0.160·39-s − 0.312·41-s + 1.21·43-s − 0.149·45-s + 0.875·47-s + 2/7·49-s − 0.140·51-s + 0.274·53-s − 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 44880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 44880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 - 12 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 - 11 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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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
−14.85875584245303, −14.36092370429156, −13.84040307303401, −13.44592485716037, −12.87630916265023, −12.25306425031280, −12.03305421711764, −11.25809449906797, −10.74340343777141, −10.03972979330355, −9.626202507222591, −9.211933049290427, −8.580062296918330, −8.062136689209424, −7.346192446417545, −7.080366723320437, −6.307050102908105, −5.780753030270164, −5.178645658663241, −4.171507040356235, −3.755218086184362, −3.391470947911478, −2.539075103324268, −1.919608028219843, −0.9115496009449915, 0,
0.9115496009449915, 1.919608028219843, 2.539075103324268, 3.391470947911478, 3.755218086184362, 4.171507040356235, 5.178645658663241, 5.780753030270164, 6.307050102908105, 7.080366723320437, 7.346192446417545, 8.062136689209424, 8.580062296918330, 9.211933049290427, 9.626202507222591, 10.03972979330355, 10.74340343777141, 11.25809449906797, 12.03305421711764, 12.25306425031280, 12.87630916265023, 13.44592485716037, 13.84040307303401, 14.36092370429156, 14.85875584245303
