| L(s) = 1 | + 2·4-s − 7-s + 3·16-s − 2·19-s + 25-s − 2·28-s + 3·31-s − 2·43-s + 49-s − 61-s + 4·64-s + 73-s − 4·76-s + 2·100-s − 3·103-s − 3·112-s + 121-s + 6·124-s + 127-s + 131-s + 2·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | + 2·4-s − 7-s + 3·16-s − 2·19-s + 25-s − 2·28-s + 3·31-s − 2·43-s + 49-s − 61-s + 4·64-s + 73-s − 4·76-s + 2·100-s − 3·103-s − 3·112-s + 121-s + 6·124-s + 127-s + 131-s + 2·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2368521 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2368521 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.702533340\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.702533340\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 5 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 7 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 23 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 31 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
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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
−10.00597349039159117736806707201, −9.601368254141669842125799212382, −9.062450808790791101198532086064, −8.413279572420646392097483091414, −8.158801950339140063273421195843, −8.049868133724637693648956409674, −7.16000461105865088925132869119, −6.93362892178194455364952346197, −6.52840832156234312111633446302, −6.46248177996568098345760344533, −5.95804595716278185437742045715, −5.57984844451973226026706801074, −4.76841214681768873303051112720, −4.48689724181073750260162206022, −3.67601876674228284170968734201, −3.29922511709881655674458067753, −2.65761141777823131694839990634, −2.55005661942751767719644105751, −1.81714122993296504725738213255, −1.09131563528078784096830501243,
1.09131563528078784096830501243, 1.81714122993296504725738213255, 2.55005661942751767719644105751, 2.65761141777823131694839990634, 3.29922511709881655674458067753, 3.67601876674228284170968734201, 4.48689724181073750260162206022, 4.76841214681768873303051112720, 5.57984844451973226026706801074, 5.95804595716278185437742045715, 6.46248177996568098345760344533, 6.52840832156234312111633446302, 6.93362892178194455364952346197, 7.16000461105865088925132869119, 8.049868133724637693648956409674, 8.158801950339140063273421195843, 8.413279572420646392097483091414, 9.062450808790791101198532086064, 9.601368254141669842125799212382, 10.00597349039159117736806707201
