| L(s) = 1 | − 2·2-s + 3·4-s + 5·7-s − 4·8-s + 4·13-s − 10·14-s + 5·16-s + 6·17-s − 2·19-s − 3·23-s + 5·25-s − 8·26-s + 15·28-s − 6·29-s + 10·31-s − 6·32-s − 12·34-s − 8·37-s + 4·38-s + 3·41-s − 2·43-s + 6·46-s + 6·47-s + 18·49-s − 10·50-s + 12·52-s + 6·53-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s + 1.88·7-s − 1.41·8-s + 1.10·13-s − 2.67·14-s + 5/4·16-s + 1.45·17-s − 0.458·19-s − 0.625·23-s + 25-s − 1.56·26-s + 2.83·28-s − 1.11·29-s + 1.79·31-s − 1.06·32-s − 2.05·34-s − 1.31·37-s + 0.648·38-s + 0.468·41-s − 0.304·43-s + 0.884·46-s + 0.875·47-s + 18/7·49-s − 1.41·50-s + 1.66·52-s + 0.824·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.807423923\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.807423923\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - 5 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 4 T + 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 3 T - 14 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 6 T + 7 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 8 T + 27 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 3 T - 32 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 2 T - 39 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 6 T - 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 11 T + 48 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 9 T - 8 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 2 T - 93 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.00762452544365622256816864013, −9.646826547508947126106558904146, −9.084912266370969053508678841913, −8.629019942972356023061823142025, −8.349843226609282130415069876291, −8.218229652425791815592869543154, −7.60498116260930810576729796684, −7.45400219865103127652679304904, −6.80832067136748689429010235194, −6.40811020488025342340455604687, −5.69775137207133593565392116214, −5.61715880863236205190427037923, −4.84657227239813815111995490214, −4.51186295546731558450092998557, −3.56212991025573369338545002227, −3.46058886128328040526502467195, −2.28677352455988310763422505300, −2.09709791278411172227671144391, −1.18571287118490488614326846276, −0.919365518272345084312670568734,
0.919365518272345084312670568734, 1.18571287118490488614326846276, 2.09709791278411172227671144391, 2.28677352455988310763422505300, 3.46058886128328040526502467195, 3.56212991025573369338545002227, 4.51186295546731558450092998557, 4.84657227239813815111995490214, 5.61715880863236205190427037923, 5.69775137207133593565392116214, 6.40811020488025342340455604687, 6.80832067136748689429010235194, 7.45400219865103127652679304904, 7.60498116260930810576729796684, 8.218229652425791815592869543154, 8.349843226609282130415069876291, 8.629019942972356023061823142025, 9.084912266370969053508678841913, 9.646826547508947126106558904146, 10.00762452544365622256816864013
