| L(s) = 1 | + 2·3-s − 2·4-s − 4·7-s + 9-s + 11-s − 4·12-s + 5·13-s + 4·16-s + 7·17-s − 8·21-s − 4·27-s + 8·28-s + 8·29-s + 4·31-s + 2·33-s − 2·36-s + 10·37-s + 10·39-s + 2·41-s + 43-s − 2·44-s − 7·47-s + 8·48-s + 9·49-s + 14·51-s − 10·52-s − 10·53-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 4-s − 1.51·7-s + 1/3·9-s + 0.301·11-s − 1.15·12-s + 1.38·13-s + 16-s + 1.69·17-s − 1.74·21-s − 0.769·27-s + 1.51·28-s + 1.48·29-s + 0.718·31-s + 0.348·33-s − 1/3·36-s + 1.64·37-s + 1.60·39-s + 0.312·41-s + 0.152·43-s − 0.301·44-s − 1.02·47-s + 1.15·48-s + 9/7·49-s + 1.96·51-s − 1.38·52-s − 1.37·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1075 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.742717868\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.742717868\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 43 | \( 1 - T \) |
| good | 2 | \( 1 + p T^{2} \) |
| 3 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 + 9 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
| 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
−9.806786808271915910137159946299, −9.026495224101588983567107354479, −8.396150677884448996555007336034, −7.72667861007177235548455450441, −6.40116964225027056733026442744, −5.75616390920891058637198920120, −4.32328180975526763288375987582, −3.40935833598822824385703242559, −2.99702509948299437244413580946, −1.01369489369341834887842904757,
1.01369489369341834887842904757, 2.99702509948299437244413580946, 3.40935833598822824385703242559, 4.32328180975526763288375987582, 5.75616390920891058637198920120, 6.40116964225027056733026442744, 7.72667861007177235548455450441, 8.396150677884448996555007336034, 9.026495224101588983567107354479, 9.806786808271915910137159946299
