| L(s) = 1 | + 3-s + 2·5-s + 3·7-s − 2·11-s − 7·13-s + 2·15-s + 4·17-s + 4·19-s + 3·21-s − 2·23-s + 3·25-s − 27-s − 6·29-s − 2·31-s − 2·33-s + 6·35-s + 6·37-s − 7·39-s − 8·41-s + 3·43-s + 20·47-s + 7·49-s + 4·51-s − 16·53-s − 4·55-s + 4·57-s − 6·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.894·5-s + 1.13·7-s − 0.603·11-s − 1.94·13-s + 0.516·15-s + 0.970·17-s + 0.917·19-s + 0.654·21-s − 0.417·23-s + 3/5·25-s − 0.192·27-s − 1.11·29-s − 0.359·31-s − 0.348·33-s + 1.01·35-s + 0.986·37-s − 1.12·39-s − 1.24·41-s + 0.457·43-s + 2.91·47-s + 49-s + 0.560·51-s − 2.19·53-s − 0.539·55-s + 0.529·57-s − 0.781·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2433600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2433600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.423830697\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.423830697\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 - T + T^{2} \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 13 | $C_2$ | \( 1 + 7 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 3 T + 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 2 T - 7 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 4 T - T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 4 T - 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 2 T - 19 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 6 T + 7 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 6 T - T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 8 T + 23 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 3 T - 34 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 6 T - 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 5 T - 36 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 3 T - 58 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 10 T + 29 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 17 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 7 T - 48 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
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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
−9.474391338148478209460890031362, −9.446406156207677013113903418001, −8.799111273365322645116460594175, −8.558719681157839570994105166259, −7.85216561340410148117207589306, −7.60296776975011636908242732021, −7.39644336358971140611089889549, −7.18316063224667413778339126708, −6.21634503811982126532004126387, −5.87649984540015327960542157268, −5.53736948859220166433823687874, −5.04405839672323883136010211420, −4.73376786514046064251913303584, −4.36084092491404173992824731001, −3.39754775789022719148205775750, −3.22161585923737291156725945848, −2.28556756004004250957589619020, −2.27591299142883469902314845600, −1.60066948247745263538326493317, −0.68962599007337531143757109900,
0.68962599007337531143757109900, 1.60066948247745263538326493317, 2.27591299142883469902314845600, 2.28556756004004250957589619020, 3.22161585923737291156725945848, 3.39754775789022719148205775750, 4.36084092491404173992824731001, 4.73376786514046064251913303584, 5.04405839672323883136010211420, 5.53736948859220166433823687874, 5.87649984540015327960542157268, 6.21634503811982126532004126387, 7.18316063224667413778339126708, 7.39644336358971140611089889549, 7.60296776975011636908242732021, 7.85216561340410148117207589306, 8.558719681157839570994105166259, 8.799111273365322645116460594175, 9.446406156207677013113903418001, 9.474391338148478209460890031362
