| L(s) = 1 | + 0.600·5-s − 7-s − 1.60·11-s + 0.330·13-s − 1.44·17-s + 2.57·19-s + 1.84·23-s − 4.63·25-s + 3.51·29-s − 9.62·31-s − 0.600·35-s + 0.600·37-s + 6.62·41-s + 3.62·43-s + 3.90·47-s + 49-s − 9.27·53-s − 0.961·55-s − 13.8·59-s − 5.18·61-s + 0.198·65-s − 11.8·67-s − 4.17·71-s + 4.13·73-s + 1.60·77-s + 8.13·79-s − 5.57·83-s + ⋯ |
| L(s) = 1 | + 0.268·5-s − 0.377·7-s − 0.482·11-s + 0.0916·13-s − 0.351·17-s + 0.591·19-s + 0.385·23-s − 0.927·25-s + 0.653·29-s − 1.72·31-s − 0.101·35-s + 0.0987·37-s + 1.03·41-s + 0.553·43-s + 0.570·47-s + 0.142·49-s − 1.27·53-s − 0.129·55-s − 1.80·59-s − 0.664·61-s + 0.0246·65-s − 1.44·67-s − 0.496·71-s + 0.484·73-s + 0.182·77-s + 0.914·79-s − 0.612·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4536 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 5 | \( 1 - 0.600T + 5T^{2} \) |
| 11 | \( 1 + 1.60T + 11T^{2} \) |
| 13 | \( 1 - 0.330T + 13T^{2} \) |
| 17 | \( 1 + 1.44T + 17T^{2} \) |
| 19 | \( 1 - 2.57T + 19T^{2} \) |
| 23 | \( 1 - 1.84T + 23T^{2} \) |
| 29 | \( 1 - 3.51T + 29T^{2} \) |
| 31 | \( 1 + 9.62T + 31T^{2} \) |
| 37 | \( 1 - 0.600T + 37T^{2} \) |
| 41 | \( 1 - 6.62T + 41T^{2} \) |
| 43 | \( 1 - 3.62T + 43T^{2} \) |
| 47 | \( 1 - 3.90T + 47T^{2} \) |
| 53 | \( 1 + 9.27T + 53T^{2} \) |
| 59 | \( 1 + 13.8T + 59T^{2} \) |
| 61 | \( 1 + 5.18T + 61T^{2} \) |
| 67 | \( 1 + 11.8T + 67T^{2} \) |
| 71 | \( 1 + 4.17T + 71T^{2} \) |
| 73 | \( 1 - 4.13T + 73T^{2} \) |
| 79 | \( 1 - 8.13T + 79T^{2} \) |
| 83 | \( 1 + 5.57T + 83T^{2} \) |
| 89 | \( 1 + 3.83T + 89T^{2} \) |
| 97 | \( 1 + 1.94T + 97T^{2} \) |
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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
−7.73439826387826853511744134818, −7.45080814084445531261889576531, −6.39232508827344027075461123047, −5.84289262136147763986092689535, −5.05863397014655954263188191520, −4.19786565141217131527440899476, −3.28388304087344188608718682487, −2.48396858276083708569067489217, −1.42281120424396556810230449314, 0,
1.42281120424396556810230449314, 2.48396858276083708569067489217, 3.28388304087344188608718682487, 4.19786565141217131527440899476, 5.05863397014655954263188191520, 5.84289262136147763986092689535, 6.39232508827344027075461123047, 7.45080814084445531261889576531, 7.73439826387826853511744134818
