| L(s) = 1 | + 3·3-s + 2·4-s + 5-s − 7-s + 6·9-s + 6·12-s + 13-s + 3·15-s − 6·17-s + 4·19-s + 2·20-s − 3·21-s − 6·23-s + 9·27-s − 2·28-s + 9·29-s + 4·31-s − 35-s + 12·36-s − 8·37-s + 3·39-s + 4·43-s + 6·45-s − 18·51-s + 2·52-s − 12·53-s + 12·57-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 4-s + 0.447·5-s − 0.377·7-s + 2·9-s + 1.73·12-s + 0.277·13-s + 0.774·15-s − 1.45·17-s + 0.917·19-s + 0.447·20-s − 0.654·21-s − 1.25·23-s + 1.73·27-s − 0.377·28-s + 1.67·29-s + 0.718·31-s − 0.169·35-s + 2·36-s − 1.31·37-s + 0.480·39-s + 0.609·43-s + 0.894·45-s − 2.52·51-s + 0.277·52-s − 1.64·53-s + 1.58·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.601310532\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.601310532\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 5 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - T - 12 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 9 T + 52 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 4 T - 27 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 6 T - 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 14 T + 129 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 9 T - 2 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 2 T - 93 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.99827690207187105144946011196, −11.35683476662989719728666577954, −10.85701809339300115835795985623, −10.49804384155786807780740293872, −9.745485071528360239938070055252, −9.693509170783262969859591377581, −9.007655204101274718009998542188, −8.665627207880056727552892041288, −7.974788613054050949889328725125, −7.85509048072689428360022057377, −6.90056040672647009729332444519, −6.78658440742861805794506094600, −6.26100813511234525757961764538, −5.54794360313809512328448145423, −4.50709596199560918723602687428, −4.24802202090610331313184639059, −3.07085494985228353924767551596, −3.03246470615294066701826701709, −2.14205569747094378523411173622, −1.62870338626646120767142495689,
1.62870338626646120767142495689, 2.14205569747094378523411173622, 3.03246470615294066701826701709, 3.07085494985228353924767551596, 4.24802202090610331313184639059, 4.50709596199560918723602687428, 5.54794360313809512328448145423, 6.26100813511234525757961764538, 6.78658440742861805794506094600, 6.90056040672647009729332444519, 7.85509048072689428360022057377, 7.974788613054050949889328725125, 8.665627207880056727552892041288, 9.007655204101274718009998542188, 9.693509170783262969859591377581, 9.745485071528360239938070055252, 10.49804384155786807780740293872, 10.85701809339300115835795985623, 11.35683476662989719728666577954, 11.99827690207187105144946011196
