L(s) = 1 | − 2.55·2-s + 0.570·3-s + 4.53·4-s − 5-s − 1.45·6-s − 4.65·7-s − 6.48·8-s − 2.67·9-s + 2.55·10-s − 11-s + 2.58·12-s + 6.22·13-s + 11.9·14-s − 0.570·15-s + 7.50·16-s − 6.53·17-s + 6.83·18-s + 8.14·19-s − 4.53·20-s − 2.65·21-s + 2.55·22-s + 0.223·23-s − 3.69·24-s + 25-s − 15.9·26-s − 3.23·27-s − 21.1·28-s + ⋯ |
L(s) = 1 | − 1.80·2-s + 0.329·3-s + 2.26·4-s − 0.447·5-s − 0.595·6-s − 1.76·7-s − 2.29·8-s − 0.891·9-s + 0.808·10-s − 0.301·11-s + 0.746·12-s + 1.72·13-s + 3.18·14-s − 0.147·15-s + 1.87·16-s − 1.58·17-s + 1.61·18-s + 1.86·19-s − 1.01·20-s − 0.579·21-s + 0.545·22-s + 0.0466·23-s − 0.754·24-s + 0.200·25-s − 3.12·26-s − 0.622·27-s − 3.99·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 73 | \( 1 + T \) |
good | 2 | \( 1 + 2.55T + 2T^{2} \) |
| 3 | \( 1 - 0.570T + 3T^{2} \) |
| 7 | \( 1 + 4.65T + 7T^{2} \) |
| 13 | \( 1 - 6.22T + 13T^{2} \) |
| 17 | \( 1 + 6.53T + 17T^{2} \) |
| 19 | \( 1 - 8.14T + 19T^{2} \) |
| 23 | \( 1 - 0.223T + 23T^{2} \) |
| 29 | \( 1 - 3.99T + 29T^{2} \) |
| 31 | \( 1 - 0.206T + 31T^{2} \) |
| 37 | \( 1 + 6.91T + 37T^{2} \) |
| 41 | \( 1 + 1.37T + 41T^{2} \) |
| 43 | \( 1 + 1.23T + 43T^{2} \) |
| 47 | \( 1 - 3.77T + 47T^{2} \) |
| 53 | \( 1 - 0.653T + 53T^{2} \) |
| 59 | \( 1 - 0.0138T + 59T^{2} \) |
| 61 | \( 1 - 13.5T + 61T^{2} \) |
| 67 | \( 1 - 11.5T + 67T^{2} \) |
| 71 | \( 1 + 14.2T + 71T^{2} \) |
| 79 | \( 1 - 3.96T + 79T^{2} \) |
| 83 | \( 1 - 14.9T + 83T^{2} \) |
| 89 | \( 1 + 8.92T + 89T^{2} \) |
| 97 | \( 1 + 13.9T + 97T^{2} \) |
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
−8.392927039314693437748855736862, −7.53014021810180211377261127077, −6.73872233818259338083587993630, −6.35981618822799517325472580499, −5.45485037997037296080289697735, −3.73643733846629664399524637004, −3.15061622741496186491365624055, −2.39515055417925269150743512336, −0.980832662544728292854431802094, 0,
0.980832662544728292854431802094, 2.39515055417925269150743512336, 3.15061622741496186491365624055, 3.73643733846629664399524637004, 5.45485037997037296080289697735, 6.35981618822799517325472580499, 6.73872233818259338083587993630, 7.53014021810180211377261127077, 8.392927039314693437748855736862