L(s) = 1 | − 2-s + 3-s + 4-s + 4.06·5-s − 6-s − 8-s − 2·9-s − 4.06·10-s − 4.67·11-s + 12-s + 1.86·13-s + 4.06·15-s + 16-s − 2.54·17-s + 2·18-s − 6.45·19-s + 4.06·20-s + 4.67·22-s − 6.32·23-s − 24-s + 11.5·25-s − 1.86·26-s − 5·27-s + 0.932·29-s − 4.06·30-s − 8.38·31-s − 32-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.81·5-s − 0.408·6-s − 0.353·8-s − 0.666·9-s − 1.28·10-s − 1.40·11-s + 0.288·12-s + 0.517·13-s + 1.04·15-s + 0.250·16-s − 0.617·17-s + 0.471·18-s − 1.48·19-s + 0.909·20-s + 0.995·22-s − 1.31·23-s − 0.204·24-s + 2.30·25-s − 0.365·26-s − 0.962·27-s + 0.173·29-s − 0.742·30-s − 1.50·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 41 | \( 1 + T \) |
good | 3 | \( 1 - T + 3T^{2} \) |
| 5 | \( 1 - 4.06T + 5T^{2} \) |
| 11 | \( 1 + 4.67T + 11T^{2} \) |
| 13 | \( 1 - 1.86T + 13T^{2} \) |
| 17 | \( 1 + 2.54T + 17T^{2} \) |
| 19 | \( 1 + 6.45T + 19T^{2} \) |
| 23 | \( 1 + 6.32T + 23T^{2} \) |
| 29 | \( 1 - 0.932T + 29T^{2} \) |
| 31 | \( 1 + 8.38T + 31T^{2} \) |
| 37 | \( 1 + 1.78T + 37T^{2} \) |
| 43 | \( 1 - 9.99T + 43T^{2} \) |
| 47 | \( 1 - 6.19T + 47T^{2} \) |
| 53 | \( 1 - 0.739T + 53T^{2} \) |
| 59 | \( 1 + 11.7T + 59T^{2} \) |
| 61 | \( 1 - 2.18T + 61T^{2} \) |
| 67 | \( 1 - 1.39T + 67T^{2} \) |
| 71 | \( 1 + 14.1T + 71T^{2} \) |
| 73 | \( 1 - 1.45T + 73T^{2} \) |
| 79 | \( 1 + 5.79T + 79T^{2} \) |
| 83 | \( 1 + 5.83T + 83T^{2} \) |
| 89 | \( 1 + 5.45T + 89T^{2} \) |
| 97 | \( 1 + 1.00T + 97T^{2} \) |
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
−8.294307674601058307909584773856, −7.53571843434323738153781011642, −6.54078194787300928521198460870, −5.83356571123113542857211591084, −5.50388376128588447215275215297, −4.22434040350359945176248244617, −2.90482381903720448687512266009, −2.29985779574087424503373167481, −1.75605606375174833472733908103, 0,
1.75605606375174833472733908103, 2.29985779574087424503373167481, 2.90482381903720448687512266009, 4.22434040350359945176248244617, 5.50388376128588447215275215297, 5.83356571123113542857211591084, 6.54078194787300928521198460870, 7.53571843434323738153781011642, 8.294307674601058307909584773856