L(s) = 1 | − 2.50·2-s − 3-s + 4.28·4-s − 2.73·5-s + 2.50·6-s − 1.54·7-s − 5.71·8-s + 9-s + 6.86·10-s − 1.30·11-s − 4.28·12-s + 3.43·13-s + 3.86·14-s + 2.73·15-s + 5.76·16-s − 2.28·17-s − 2.50·18-s − 2.39·19-s − 11.7·20-s + 1.54·21-s + 3.27·22-s + 23-s + 5.71·24-s + 2.50·25-s − 8.60·26-s − 27-s − 6.59·28-s + ⋯ |
L(s) = 1 | − 1.77·2-s − 0.577·3-s + 2.14·4-s − 1.22·5-s + 1.02·6-s − 0.582·7-s − 2.02·8-s + 0.333·9-s + 2.17·10-s − 0.393·11-s − 1.23·12-s + 0.952·13-s + 1.03·14-s + 0.707·15-s + 1.44·16-s − 0.553·17-s − 0.590·18-s − 0.548·19-s − 2.62·20-s + 0.336·21-s + 0.697·22-s + 0.208·23-s + 1.16·24-s + 0.501·25-s − 1.68·26-s − 0.192·27-s − 1.24·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
good | 2 | \( 1 + 2.50T + 2T^{2} \) |
| 5 | \( 1 + 2.73T + 5T^{2} \) |
| 7 | \( 1 + 1.54T + 7T^{2} \) |
| 11 | \( 1 + 1.30T + 11T^{2} \) |
| 13 | \( 1 - 3.43T + 13T^{2} \) |
| 17 | \( 1 + 2.28T + 17T^{2} \) |
| 19 | \( 1 + 2.39T + 19T^{2} \) |
| 31 | \( 1 - 2.47T + 31T^{2} \) |
| 37 | \( 1 - 10.3T + 37T^{2} \) |
| 41 | \( 1 + 3.81T + 41T^{2} \) |
| 43 | \( 1 - 4.29T + 43T^{2} \) |
| 47 | \( 1 - 10.4T + 47T^{2} \) |
| 53 | \( 1 + 0.703T + 53T^{2} \) |
| 59 | \( 1 + 14.9T + 59T^{2} \) |
| 61 | \( 1 + 0.0657T + 61T^{2} \) |
| 67 | \( 1 + 0.500T + 67T^{2} \) |
| 71 | \( 1 + 0.422T + 71T^{2} \) |
| 73 | \( 1 - 11.1T + 73T^{2} \) |
| 79 | \( 1 + 2.63T + 79T^{2} \) |
| 83 | \( 1 - 15.9T + 83T^{2} \) |
| 89 | \( 1 - 15.8T + 89T^{2} \) |
| 97 | \( 1 - 3.71T + 97T^{2} \) |
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
−8.806014209515192850562666857668, −7.954565283564172651526302788148, −7.56367475864801142323470437384, −6.55529144143285644056407892779, −6.12633247855376382650470621356, −4.62922247106009666986247257325, −3.61509556081520773657488788115, −2.45111046656329092446787127668, −0.994633543089141866606701045434, 0,
0.994633543089141866606701045434, 2.45111046656329092446787127668, 3.61509556081520773657488788115, 4.62922247106009666986247257325, 6.12633247855376382650470621356, 6.55529144143285644056407892779, 7.56367475864801142323470437384, 7.954565283564172651526302788148, 8.806014209515192850562666857668