| L(s) = 1 | + 3-s − 2.34·5-s − 7-s + 9-s + 2.85·11-s − 0.518·13-s − 2.34·15-s − 2.17·17-s − 1.14·19-s − 21-s + 23-s + 0.481·25-s + 27-s − 2.17·29-s + 5.89·31-s + 2.85·33-s + 2.34·35-s − 9.89·37-s − 0.518·39-s + 6.85·41-s + 2.23·43-s − 2.34·45-s − 5.03·47-s + 49-s − 2.17·51-s − 1.30·53-s − 6.69·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.04·5-s − 0.377·7-s + 0.333·9-s + 0.862·11-s − 0.143·13-s − 0.604·15-s − 0.527·17-s − 0.261·19-s − 0.218·21-s + 0.208·23-s + 0.0963·25-s + 0.192·27-s − 0.404·29-s + 1.05·31-s + 0.497·33-s + 0.395·35-s − 1.62·37-s − 0.0829·39-s + 1.07·41-s + 0.341·43-s − 0.349·45-s − 0.734·47-s + 0.142·49-s − 0.304·51-s − 0.179·53-s − 0.902·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 + 2.34T + 5T^{2} \) |
| 11 | \( 1 - 2.85T + 11T^{2} \) |
| 13 | \( 1 + 0.518T + 13T^{2} \) |
| 17 | \( 1 + 2.17T + 17T^{2} \) |
| 19 | \( 1 + 1.14T + 19T^{2} \) |
| 29 | \( 1 + 2.17T + 29T^{2} \) |
| 31 | \( 1 - 5.89T + 31T^{2} \) |
| 37 | \( 1 + 9.89T + 37T^{2} \) |
| 41 | \( 1 - 6.85T + 41T^{2} \) |
| 43 | \( 1 - 2.23T + 43T^{2} \) |
| 47 | \( 1 + 5.03T + 47T^{2} \) |
| 53 | \( 1 + 1.30T + 53T^{2} \) |
| 59 | \( 1 + 8.41T + 59T^{2} \) |
| 61 | \( 1 + 6.16T + 61T^{2} \) |
| 67 | \( 1 + 14.4T + 67T^{2} \) |
| 71 | \( 1 - 2.51T + 71T^{2} \) |
| 73 | \( 1 - 7.21T + 73T^{2} \) |
| 79 | \( 1 + 8.85T + 79T^{2} \) |
| 83 | \( 1 - 12.5T + 83T^{2} \) |
| 89 | \( 1 + 14.5T + 89T^{2} \) |
| 97 | \( 1 - 1.82T + 97T^{2} \) |
| 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
−8.093715837337372526674876227338, −7.46843508231272182759531332553, −6.76198466517805387755456818514, −6.07563832067743967213070055747, −4.84288898971428583755625781255, −4.13211505178256567157006554194, −3.51773287067907128418798778876, −2.64844926765797918249999603555, −1.44396882014989010030824755159, 0,
1.44396882014989010030824755159, 2.64844926765797918249999603555, 3.51773287067907128418798778876, 4.13211505178256567157006554194, 4.84288898971428583755625781255, 6.07563832067743967213070055747, 6.76198466517805387755456818514, 7.46843508231272182759531332553, 8.093715837337372526674876227338
