| L(s) = 1 | − 1.47·5-s + 2.64·7-s + 6.34·11-s − 5.34·13-s + 0.715·17-s + 3.28·19-s − 0.885·23-s − 2.83·25-s + 0.885·29-s + 7.47·31-s − 3.89·35-s − 37-s + 7.75·41-s + 10.1·43-s − 11.8·47-s − 0.0194·49-s − 4.15·53-s − 9.34·55-s + 12.1·59-s + 3.81·61-s + 7.87·65-s − 5.32·67-s − 2.87·71-s − 8.34·73-s + 16.7·77-s − 1.83·79-s − 2.15·83-s + ⋯ |
| L(s) = 1 | − 0.658·5-s + 0.998·7-s + 1.91·11-s − 1.48·13-s + 0.173·17-s + 0.753·19-s − 0.184·23-s − 0.566·25-s + 0.164·29-s + 1.34·31-s − 0.657·35-s − 0.164·37-s + 1.21·41-s + 1.54·43-s − 1.72·47-s − 0.00277·49-s − 0.570·53-s − 1.26·55-s + 1.57·59-s + 0.487·61-s + 0.976·65-s − 0.650·67-s − 0.340·71-s − 0.976·73-s + 1.91·77-s − 0.205·79-s − 0.236·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2664 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.947269172\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.947269172\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 37 | \( 1 + T \) |
| good | 5 | \( 1 + 1.47T + 5T^{2} \) |
| 7 | \( 1 - 2.64T + 7T^{2} \) |
| 11 | \( 1 - 6.34T + 11T^{2} \) |
| 13 | \( 1 + 5.34T + 13T^{2} \) |
| 17 | \( 1 - 0.715T + 17T^{2} \) |
| 19 | \( 1 - 3.28T + 19T^{2} \) |
| 23 | \( 1 + 0.885T + 23T^{2} \) |
| 29 | \( 1 - 0.885T + 29T^{2} \) |
| 31 | \( 1 - 7.47T + 31T^{2} \) |
| 41 | \( 1 - 7.75T + 41T^{2} \) |
| 43 | \( 1 - 10.1T + 43T^{2} \) |
| 47 | \( 1 + 11.8T + 47T^{2} \) |
| 53 | \( 1 + 4.15T + 53T^{2} \) |
| 59 | \( 1 - 12.1T + 59T^{2} \) |
| 61 | \( 1 - 3.81T + 61T^{2} \) |
| 67 | \( 1 + 5.32T + 67T^{2} \) |
| 71 | \( 1 + 2.87T + 71T^{2} \) |
| 73 | \( 1 + 8.34T + 73T^{2} \) |
| 79 | \( 1 + 1.83T + 79T^{2} \) |
| 83 | \( 1 + 2.15T + 83T^{2} \) |
| 89 | \( 1 - 1.89T + 89T^{2} \) |
| 97 | \( 1 - 15.0T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.845621503438382224400779047078, −7.959253477440208852477444383680, −7.46831117319741735603407535178, −6.67920843118123391739510750602, −5.75402553305669644203269000236, −4.67493027157922854612594253043, −4.26914812625389816914390275517, −3.23147843869033772616768329954, −2.01514899171345556991368943836, −0.917300351310702402452691260785,
0.917300351310702402452691260785, 2.01514899171345556991368943836, 3.23147843869033772616768329954, 4.26914812625389816914390275517, 4.67493027157922854612594253043, 5.75402553305669644203269000236, 6.67920843118123391739510750602, 7.46831117319741735603407535178, 7.959253477440208852477444383680, 8.845621503438382224400779047078
