| L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s − 7-s − 8-s + 9-s − 10-s + 11-s − 12-s + 13-s + 14-s − 15-s + 16-s − 17-s − 18-s − 19-s + 20-s + 21-s − 22-s + 23-s + 24-s + 25-s − 26-s − 27-s − 28-s + ⋯ |
| L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s − 7-s − 8-s + 9-s − 10-s + 11-s − 12-s + 13-s + 14-s − 15-s + 16-s − 17-s − 18-s − 19-s + 20-s + 21-s − 22-s + 23-s + 24-s + 25-s − 26-s − 27-s − 28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.074272556\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.074272556\) |
| \(L(1)\) |
\(\approx\) |
\(0.6573578036\) |
| \(L(1)\) |
\(\approx\) |
\(0.6573578036\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 571 | \( 1 \) |
| good | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| 31 | \( 1 + T \) |
| 37 | \( 1 + T \) |
| 41 | \( 1 - T \) |
| 43 | \( 1 + T \) |
| 47 | \( 1 - T \) |
| 53 | \( 1 - T \) |
| 59 | \( 1 + T \) |
| 61 | \( 1 + T \) |
| 67 | \( 1 - T \) |
| 71 | \( 1 - T \) |
| 73 | \( 1 - T \) |
| 79 | \( 1 - T \) |
| 83 | \( 1 + T \) |
| 89 | \( 1 - T \) |
| 97 | \( 1 + T \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−22.99146055773987900730461525328, −22.126982908783400371368706974175, −21.41018960292468386472637323984, −20.561043451437356453282984802115, −19.377129754936802606108311060920, −18.82318177770295959997742000968, −17.737356770377053913814853657329, −17.364905712872264496803286878541, −16.52080499922712223826126004251, −15.89153635077278392917199748919, −14.874610707705053723610514582916, −13.40971946145663908242863859126, −12.74760507805308844991762815124, −11.66484440980366309121598034602, −10.83141329981147414415254579522, −10.11354738597136964924114120905, −9.2974693867762719145691016809, −8.559763107029195680084275556, −6.76772858689047143121137669057, −6.54793069014065520862697595562, −5.82861763861029180616296547573, −4.3370392690319550049593588405, −2.890272046017520778664080100120, −1.61060305098071281966869382743, −0.6977113621536256661018854704,
0.6977113621536256661018854704, 1.61060305098071281966869382743, 2.890272046017520778664080100120, 4.3370392690319550049593588405, 5.82861763861029180616296547573, 6.54793069014065520862697595562, 6.76772858689047143121137669057, 8.559763107029195680084275556, 9.2974693867762719145691016809, 10.11354738597136964924114120905, 10.83141329981147414415254579522, 11.66484440980366309121598034602, 12.74760507805308844991762815124, 13.40971946145663908242863859126, 14.874610707705053723610514582916, 15.89153635077278392917199748919, 16.52080499922712223826126004251, 17.364905712872264496803286878541, 17.737356770377053913814853657329, 18.82318177770295959997742000968, 19.377129754936802606108311060920, 20.561043451437356453282984802115, 21.41018960292468386472637323984, 22.126982908783400371368706974175, 22.99146055773987900730461525328
