| L(s) = 1 | − 2-s + 4-s − 1.41·5-s − 8-s − 3·9-s + 1.41·10-s + 5.65·11-s − 4·13-s + 16-s + 3·18-s − 4·19-s − 1.41·20-s − 5.65·22-s − 5.65·23-s − 2.99·25-s + 4·26-s − 4.24·29-s + 5.65·31-s − 32-s − 3·36-s − 4.24·37-s + 4·38-s + 1.41·40-s + 1.41·41-s − 4·43-s + 5.65·44-s + 4.24·45-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.632·5-s − 0.353·8-s − 9-s + 0.447·10-s + 1.70·11-s − 1.10·13-s + 0.250·16-s + 0.707·18-s − 0.917·19-s − 0.316·20-s − 1.20·22-s − 1.17·23-s − 0.599·25-s + 0.784·26-s − 0.787·29-s + 1.01·31-s − 0.176·32-s − 0.5·36-s − 0.697·37-s + 0.648·38-s + 0.223·40-s + 0.220·41-s − 0.609·43-s + 0.852·44-s + 0.632·45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 578 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 578 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 3 | \( 1 + 3T^{2} \) |
| 5 | \( 1 + 1.41T + 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 - 5.65T + 11T^{2} \) |
| 13 | \( 1 + 4T + 13T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 + 5.65T + 23T^{2} \) |
| 29 | \( 1 + 4.24T + 29T^{2} \) |
| 31 | \( 1 - 5.65T + 31T^{2} \) |
| 37 | \( 1 + 4.24T + 37T^{2} \) |
| 41 | \( 1 - 1.41T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 + 8T + 47T^{2} \) |
| 53 | \( 1 - 4T + 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 + 12.7T + 61T^{2} \) |
| 67 | \( 1 + 12T + 67T^{2} \) |
| 71 | \( 1 - 5.65T + 71T^{2} \) |
| 73 | \( 1 - 7.07T + 73T^{2} \) |
| 79 | \( 1 + 11.3T + 79T^{2} \) |
| 83 | \( 1 - 4T + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 4.24T + 97T^{2} \) |
| 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
−10.13784022281610075984313966331, −9.352439858591010014084835862609, −8.523215776568282963554455033595, −7.79779479805580298688913788128, −6.74101254896308828181369964479, −5.95880979232708495560048375951, −4.48583644551997754296942229128, −3.39469334072904059378445122103, −1.94947439300087123984513351010, 0,
1.94947439300087123984513351010, 3.39469334072904059378445122103, 4.48583644551997754296942229128, 5.95880979232708495560048375951, 6.74101254896308828181369964479, 7.79779479805580298688913788128, 8.523215776568282963554455033595, 9.352439858591010014084835862609, 10.13784022281610075984313966331
