L(s) = 1 | − 1.61·2-s + 1.23·3-s + 0.618·4-s − 5-s − 2.00·6-s − 4.61·7-s + 2.23·8-s − 1.47·9-s + 1.61·10-s − 11-s + 0.763·12-s − 2.61·13-s + 7.47·14-s − 1.23·15-s − 4.85·16-s − 5.09·17-s + 2.38·18-s − 5·19-s − 0.618·20-s − 5.70·21-s + 1.61·22-s − 6.61·23-s + 2.76·24-s − 4·25-s + 4.23·26-s − 5.52·27-s − 2.85·28-s + ⋯ |
L(s) = 1 | − 1.14·2-s + 0.713·3-s + 0.309·4-s − 0.447·5-s − 0.816·6-s − 1.74·7-s + 0.790·8-s − 0.490·9-s + 0.511·10-s − 0.301·11-s + 0.220·12-s − 0.726·13-s + 1.99·14-s − 0.319·15-s − 1.21·16-s − 1.23·17-s + 0.561·18-s − 1.14·19-s − 0.138·20-s − 1.24·21-s + 0.344·22-s − 1.37·23-s + 0.564·24-s − 0.800·25-s + 0.830·26-s − 1.06·27-s − 0.539·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 4027 | \( 1+O(T) \) |
good | 2 | \( 1 + 1.61T + 2T^{2} \) |
| 3 | \( 1 - 1.23T + 3T^{2} \) |
| 5 | \( 1 + T + 5T^{2} \) |
| 7 | \( 1 + 4.61T + 7T^{2} \) |
| 11 | \( 1 + T + 11T^{2} \) |
| 13 | \( 1 + 2.61T + 13T^{2} \) |
| 17 | \( 1 + 5.09T + 17T^{2} \) |
| 19 | \( 1 + 5T + 19T^{2} \) |
| 23 | \( 1 + 6.61T + 23T^{2} \) |
| 29 | \( 1 + 3.38T + 29T^{2} \) |
| 31 | \( 1 + 3.76T + 31T^{2} \) |
| 37 | \( 1 - 6.94T + 37T^{2} \) |
| 41 | \( 1 + 2.76T + 41T^{2} \) |
| 43 | \( 1 - 8.70T + 43T^{2} \) |
| 47 | \( 1 - 0.472T + 47T^{2} \) |
| 53 | \( 1 - 1.76T + 53T^{2} \) |
| 59 | \( 1 - 4.70T + 59T^{2} \) |
| 61 | \( 1 + 11.1T + 61T^{2} \) |
| 67 | \( 1 + 5.70T + 67T^{2} \) |
| 71 | \( 1 + 4.14T + 71T^{2} \) |
| 73 | \( 1 - 0.0901T + 73T^{2} \) |
| 79 | \( 1 - 13T + 79T^{2} \) |
| 83 | \( 1 + 16.5T + 83T^{2} \) |
| 89 | \( 1 + 4.14T + 89T^{2} \) |
| 97 | \( 1 - 4.52T + 97T^{2} \) |
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
−7.85224436871483197335649174826, −7.30974896975348319763846171450, −6.44088221649436062100150352183, −5.74584922460636842827468475644, −4.31717581832264436454681995722, −3.82404021946333422453026596635, −2.67061473860151061007137077991, −2.09389044360463292709620624657, 0, 0,
2.09389044360463292709620624657, 2.67061473860151061007137077991, 3.82404021946333422453026596635, 4.31717581832264436454681995722, 5.74584922460636842827468475644, 6.44088221649436062100150352183, 7.30974896975348319763846171450, 7.85224436871483197335649174826