L(s) = 1 | − 2-s − 3·3-s + 4-s − 4·5-s + 3·6-s − 5·7-s − 8-s + 6·9-s + 4·10-s − 6·11-s − 3·12-s − 6·13-s + 5·14-s + 12·15-s + 16-s − 6·17-s − 6·18-s − 8·19-s − 4·20-s + 15·21-s + 6·22-s − 6·23-s + 3·24-s + 11·25-s + 6·26-s − 9·27-s − 5·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.73·3-s + 1/2·4-s − 1.78·5-s + 1.22·6-s − 1.88·7-s − 0.353·8-s + 2·9-s + 1.26·10-s − 1.80·11-s − 0.866·12-s − 1.66·13-s + 1.33·14-s + 3.09·15-s + 1/4·16-s − 1.45·17-s − 1.41·18-s − 1.83·19-s − 0.894·20-s + 3.27·21-s + 1.27·22-s − 1.25·23-s + 0.612·24-s + 11/5·25-s + 1.17·26-s − 1.73·27-s − 0.944·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 234446 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 234446 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 117223 | \( 1 - T \) |
good | 3 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 + 4 T + p T^{2} \) |
| 7 | \( 1 + 5 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 11 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 + 13 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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
−13.33368034086200, −13.09360406818600, −12.73824472819508, −12.24614205522120, −12.04497230225563, −11.55674326486266, −10.89126786866145, −10.58503012146236, −10.35721937854419, −9.879771989628829, −9.261689397031543, −8.645433260663214, −8.136132709760610, −7.500600983907722, −7.222532857430923, −6.730090190731062, −6.443618701646912, −5.825068468989255, −5.165585013784794, −4.717665714334101, −4.254722279811606, −3.589684506825036, −3.045158921575872, −2.335210734225171, −1.733532247157827, 0, 0, 0, 0,
1.733532247157827, 2.335210734225171, 3.045158921575872, 3.589684506825036, 4.254722279811606, 4.717665714334101, 5.165585013784794, 5.825068468989255, 6.443618701646912, 6.730090190731062, 7.222532857430923, 7.500600983907722, 8.136132709760610, 8.645433260663214, 9.261689397031543, 9.879771989628829, 10.35721937854419, 10.58503012146236, 10.89126786866145, 11.55674326486266, 12.04497230225563, 12.24614205522120, 12.73824472819508, 13.09360406818600, 13.33368034086200