L(s) = 1 | + 2.62·2-s − 0.928·3-s + 4.87·4-s − 5-s − 2.43·6-s − 3.83·7-s + 7.54·8-s − 2.13·9-s − 2.62·10-s − 3.26·11-s − 4.52·12-s − 0.364·13-s − 10.0·14-s + 0.928·15-s + 10.0·16-s − 0.684·17-s − 5.60·18-s − 4.87·20-s + 3.55·21-s − 8.56·22-s − 9.36·23-s − 7.00·24-s + 25-s − 0.954·26-s + 4.77·27-s − 18.6·28-s + 1.53·29-s + ⋯ |
L(s) = 1 | + 1.85·2-s − 0.536·3-s + 2.43·4-s − 0.447·5-s − 0.994·6-s − 1.44·7-s + 2.66·8-s − 0.712·9-s − 0.829·10-s − 0.985·11-s − 1.30·12-s − 0.100·13-s − 2.68·14-s + 0.239·15-s + 2.50·16-s − 0.165·17-s − 1.32·18-s − 1.09·20-s + 0.776·21-s − 1.82·22-s − 1.95·23-s − 1.42·24-s + 0.200·25-s − 0.187·26-s + 0.918·27-s − 3.53·28-s + 0.285·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 19 | \( 1 \) |
good | 2 | \( 1 - 2.62T + 2T^{2} \) |
| 3 | \( 1 + 0.928T + 3T^{2} \) |
| 7 | \( 1 + 3.83T + 7T^{2} \) |
| 11 | \( 1 + 3.26T + 11T^{2} \) |
| 13 | \( 1 + 0.364T + 13T^{2} \) |
| 17 | \( 1 + 0.684T + 17T^{2} \) |
| 23 | \( 1 + 9.36T + 23T^{2} \) |
| 29 | \( 1 - 1.53T + 29T^{2} \) |
| 31 | \( 1 + 2.44T + 31T^{2} \) |
| 37 | \( 1 - 0.163T + 37T^{2} \) |
| 41 | \( 1 + 7.37T + 41T^{2} \) |
| 43 | \( 1 - 2.58T + 43T^{2} \) |
| 47 | \( 1 - 8.79T + 47T^{2} \) |
| 53 | \( 1 - 11.1T + 53T^{2} \) |
| 59 | \( 1 + 7.78T + 59T^{2} \) |
| 61 | \( 1 - 2.41T + 61T^{2} \) |
| 67 | \( 1 + 9.55T + 67T^{2} \) |
| 71 | \( 1 + 5.12T + 71T^{2} \) |
| 73 | \( 1 + 7.88T + 73T^{2} \) |
| 79 | \( 1 + 9.32T + 79T^{2} \) |
| 83 | \( 1 - 0.729T + 83T^{2} \) |
| 89 | \( 1 - 12.6T + 89T^{2} \) |
| 97 | \( 1 + 7.54T + 97T^{2} \) |
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
−8.783499835223777633426298730566, −7.67295539213433502940224259342, −6.93960702924550343661921804465, −5.99711294919372441544580679373, −5.79758209265530482286314978517, −4.77037317854026560524039496588, −3.87286781261623783483957892016, −3.10319389933574561654846941743, −2.34078253538495758748695285855, 0,
2.34078253538495758748695285855, 3.10319389933574561654846941743, 3.87286781261623783483957892016, 4.77037317854026560524039496588, 5.79758209265530482286314978517, 5.99711294919372441544580679373, 6.93960702924550343661921804465, 7.67295539213433502940224259342, 8.783499835223777633426298730566