L(s) = 1 | + 0.949·2-s − 2.99·3-s − 1.09·4-s − 0.667·5-s − 2.84·6-s − 3.07·7-s − 2.94·8-s + 5.96·9-s − 0.633·10-s + 0.288·11-s + 3.29·12-s − 5.99·13-s − 2.92·14-s + 1.99·15-s − 0.594·16-s + 2.27·17-s + 5.66·18-s + 6.30·19-s + 0.733·20-s + 9.21·21-s + 0.273·22-s + 6.81·23-s + 8.80·24-s − 4.55·25-s − 5.69·26-s − 8.87·27-s + 3.38·28-s + ⋯ |
L(s) = 1 | + 0.671·2-s − 1.72·3-s − 0.549·4-s − 0.298·5-s − 1.16·6-s − 1.16·7-s − 1.04·8-s + 1.98·9-s − 0.200·10-s + 0.0869·11-s + 0.949·12-s − 1.66·13-s − 0.780·14-s + 0.515·15-s − 0.148·16-s + 0.550·17-s + 1.33·18-s + 1.44·19-s + 0.163·20-s + 2.01·21-s + 0.0583·22-s + 1.42·23-s + 1.79·24-s − 0.910·25-s − 1.11·26-s − 1.70·27-s + 0.639·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 4001 | \( 1+O(T) \) |
good | 2 | \( 1 - 0.949T + 2T^{2} \) |
| 3 | \( 1 + 2.99T + 3T^{2} \) |
| 5 | \( 1 + 0.667T + 5T^{2} \) |
| 7 | \( 1 + 3.07T + 7T^{2} \) |
| 11 | \( 1 - 0.288T + 11T^{2} \) |
| 13 | \( 1 + 5.99T + 13T^{2} \) |
| 17 | \( 1 - 2.27T + 17T^{2} \) |
| 19 | \( 1 - 6.30T + 19T^{2} \) |
| 23 | \( 1 - 6.81T + 23T^{2} \) |
| 29 | \( 1 - 3.07T + 29T^{2} \) |
| 31 | \( 1 + 5.87T + 31T^{2} \) |
| 37 | \( 1 - 8.77T + 37T^{2} \) |
| 41 | \( 1 + 2.99T + 41T^{2} \) |
| 43 | \( 1 - 3.17T + 43T^{2} \) |
| 47 | \( 1 - 2.56T + 47T^{2} \) |
| 53 | \( 1 - 10.2T + 53T^{2} \) |
| 59 | \( 1 + 12.9T + 59T^{2} \) |
| 61 | \( 1 + 11.7T + 61T^{2} \) |
| 67 | \( 1 - 7.70T + 67T^{2} \) |
| 71 | \( 1 + 3.96T + 71T^{2} \) |
| 73 | \( 1 - 10.9T + 73T^{2} \) |
| 79 | \( 1 - 2.13T + 79T^{2} \) |
| 83 | \( 1 + 6.40T + 83T^{2} \) |
| 89 | \( 1 - 1.99T + 89T^{2} \) |
| 97 | \( 1 - 17.5T + 97T^{2} \) |
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
−7.67473815366973575726283916870, −7.17919695805143424169643616085, −6.35740255188829622224223714203, −5.68514794737940019738583363804, −5.13614878452757173107460771847, −4.56549440987262855434385260346, −3.62122122326916420415535508258, −2.80162070950751674877516644194, −0.895843755082532398222088854785, 0,
0.895843755082532398222088854785, 2.80162070950751674877516644194, 3.62122122326916420415535508258, 4.56549440987262855434385260346, 5.13614878452757173107460771847, 5.68514794737940019738583363804, 6.35740255188829622224223714203, 7.17919695805143424169643616085, 7.67473815366973575726283916870