| L(s) = 1 | − 1.12·3-s − 4.42·5-s − 1.73·9-s + 11-s − 5.85·13-s + 4.98·15-s + 1.55·17-s + 7.40·19-s + 4.98·23-s + 14.5·25-s + 5.32·27-s − 9.10·29-s + 1.43·31-s − 1.12·33-s − 0.731·37-s + 6.59·39-s + 1.55·41-s + 2.59·43-s + 7.66·45-s + 0.953·47-s − 1.74·51-s + 2·53-s − 4.42·55-s − 8.34·57-s + 8.22·59-s − 3.60·61-s + 25.9·65-s + ⋯ |
| L(s) = 1 | − 0.650·3-s − 1.97·5-s − 0.577·9-s + 0.301·11-s − 1.62·13-s + 1.28·15-s + 0.376·17-s + 1.69·19-s + 1.03·23-s + 2.91·25-s + 1.02·27-s − 1.69·29-s + 0.257·31-s − 0.196·33-s − 0.120·37-s + 1.05·39-s + 0.242·41-s + 0.396·43-s + 1.14·45-s + 0.139·47-s − 0.244·51-s + 0.274·53-s − 0.596·55-s − 1.10·57-s + 1.07·59-s − 0.461·61-s + 3.21·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 + 1.12T + 3T^{2} \) |
| 5 | \( 1 + 4.42T + 5T^{2} \) |
| 13 | \( 1 + 5.85T + 13T^{2} \) |
| 17 | \( 1 - 1.55T + 17T^{2} \) |
| 19 | \( 1 - 7.40T + 19T^{2} \) |
| 23 | \( 1 - 4.98T + 23T^{2} \) |
| 29 | \( 1 + 9.10T + 29T^{2} \) |
| 31 | \( 1 - 1.43T + 31T^{2} \) |
| 37 | \( 1 + 0.731T + 37T^{2} \) |
| 41 | \( 1 - 1.55T + 41T^{2} \) |
| 43 | \( 1 - 2.59T + 43T^{2} \) |
| 47 | \( 1 - 0.953T + 47T^{2} \) |
| 53 | \( 1 - 2T + 53T^{2} \) |
| 59 | \( 1 - 8.22T + 59T^{2} \) |
| 61 | \( 1 + 3.60T + 61T^{2} \) |
| 67 | \( 1 - 9.32T + 67T^{2} \) |
| 71 | \( 1 + 0.478T + 71T^{2} \) |
| 73 | \( 1 + 6.16T + 73T^{2} \) |
| 79 | \( 1 + 9.74T + 79T^{2} \) |
| 83 | \( 1 + 10.5T + 83T^{2} \) |
| 89 | \( 1 + 4.11T + 89T^{2} \) |
| 97 | \( 1 - 11.3T + 97T^{2} \) |
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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
−7.76674760354898479754707931632, −7.35265965256166924974454612777, −6.86149084669862656266958079870, −5.54178359297263811182093024504, −5.08361281243071185437574238847, −4.27371561941485666193126978200, −3.39536750035770875024921048060, −2.75018371039450776551633769324, −0.940940477612996827932859266641, 0,
0.940940477612996827932859266641, 2.75018371039450776551633769324, 3.39536750035770875024921048060, 4.27371561941485666193126978200, 5.08361281243071185437574238847, 5.54178359297263811182093024504, 6.86149084669862656266958079870, 7.35265965256166924974454612777, 7.76674760354898479754707931632
