L(s) = 1 | + 2-s + 3-s + 4-s − 5-s + 6-s + 7-s + 8-s − 2·9-s − 10-s − 3·11-s + 12-s + 5·13-s + 14-s − 15-s + 16-s − 6·17-s − 2·18-s + 4·19-s − 20-s + 21-s − 3·22-s + 23-s + 24-s − 4·25-s + 5·26-s − 5·27-s + 28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s − 2/3·9-s − 0.316·10-s − 0.904·11-s + 0.288·12-s + 1.38·13-s + 0.267·14-s − 0.258·15-s + 1/4·16-s − 1.45·17-s − 0.471·18-s + 0.917·19-s − 0.223·20-s + 0.218·21-s − 0.639·22-s + 0.208·23-s + 0.204·24-s − 4/5·25-s + 0.980·26-s − 0.962·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 270802 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 270802 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 \) |
good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 11 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 - 5 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 - 15 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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
−13.25379583620796, −12.71335502242285, −11.98020471182882, −11.49113858163370, −11.37203418553526, −10.94443512210117, −10.33321635091582, −9.793912372095578, −9.252202155700878, −8.677589270876650, −8.164710912028489, −8.034262795208281, −7.544998525448476, −6.738140188264107, −6.360056557544974, −5.920025990034816, −5.276101088321707, −4.856374703996802, −4.247553297418412, −3.856468222557575, −3.173710735305664, −2.811945129858950, −2.304396310942473, −1.610564956388407, −0.8764837228654637, 0,
0.8764837228654637, 1.610564956388407, 2.304396310942473, 2.811945129858950, 3.173710735305664, 3.856468222557575, 4.247553297418412, 4.856374703996802, 5.276101088321707, 5.920025990034816, 6.360056557544974, 6.738140188264107, 7.544998525448476, 8.034262795208281, 8.164710912028489, 8.677589270876650, 9.252202155700878, 9.793912372095578, 10.33321635091582, 10.94443512210117, 11.37203418553526, 11.49113858163370, 11.98020471182882, 12.71335502242285, 13.25379583620796