| L(s) = 1 | + 2-s + 0.732·3-s + 4-s − 5-s + 0.732·6-s − 7-s + 8-s − 2.46·9-s − 10-s − 0.732·11-s + 0.732·12-s − 14-s − 0.732·15-s + 16-s − 5.73·17-s − 2.46·18-s + 1.46·19-s − 20-s − 0.732·21-s − 0.732·22-s + 1.26·23-s + 0.732·24-s − 4·25-s − 4·27-s − 28-s − 3·29-s − 0.732·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.422·3-s + 0.5·4-s − 0.447·5-s + 0.298·6-s − 0.377·7-s + 0.353·8-s − 0.821·9-s − 0.316·10-s − 0.220·11-s + 0.211·12-s − 0.267·14-s − 0.189·15-s + 0.250·16-s − 1.39·17-s − 0.580·18-s + 0.335·19-s − 0.223·20-s − 0.159·21-s − 0.156·22-s + 0.264·23-s + 0.149·24-s − 0.800·25-s − 0.769·27-s − 0.188·28-s − 0.557·29-s − 0.133·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2366 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2366 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - 0.732T + 3T^{2} \) |
| 5 | \( 1 + T + 5T^{2} \) |
| 11 | \( 1 + 0.732T + 11T^{2} \) |
| 17 | \( 1 + 5.73T + 17T^{2} \) |
| 19 | \( 1 - 1.46T + 19T^{2} \) |
| 23 | \( 1 - 1.26T + 23T^{2} \) |
| 29 | \( 1 + 3T + 29T^{2} \) |
| 31 | \( 1 - 5.26T + 31T^{2} \) |
| 37 | \( 1 + 5.19T + 37T^{2} \) |
| 41 | \( 1 + 2.46T + 41T^{2} \) |
| 43 | \( 1 + 12.1T + 43T^{2} \) |
| 47 | \( 1 - 2.92T + 47T^{2} \) |
| 53 | \( 1 - 1.53T + 53T^{2} \) |
| 59 | \( 1 + 10.7T + 59T^{2} \) |
| 61 | \( 1 + 11.7T + 61T^{2} \) |
| 67 | \( 1 - 11.6T + 67T^{2} \) |
| 71 | \( 1 - 13.8T + 71T^{2} \) |
| 73 | \( 1 + 11.3T + 73T^{2} \) |
| 79 | \( 1 + 3.80T + 79T^{2} \) |
| 83 | \( 1 + 3.80T + 83T^{2} \) |
| 89 | \( 1 + 2.53T + 89T^{2} \) |
| 97 | \( 1 - 5.46T + 97T^{2} \) |
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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
−8.515555040556149667672946354483, −7.84023186710153715636150745632, −6.94889281087645164076829857500, −6.25021980417363742317247007268, −5.36956929014345924273026435746, −4.52245195760403103912932975837, −3.59619903384691236174980648320, −2.91204849639759265843545105556, −1.93566937048094380354569967793, 0,
1.93566937048094380354569967793, 2.91204849639759265843545105556, 3.59619903384691236174980648320, 4.52245195760403103912932975837, 5.36956929014345924273026435746, 6.25021980417363742317247007268, 6.94889281087645164076829857500, 7.84023186710153715636150745632, 8.515555040556149667672946354483
