L(s) = 1 | + 2·3-s + 5-s − 6·7-s + 3·9-s + 10·11-s − 4·13-s + 2·15-s + 3·17-s − 12·21-s − 8·23-s + 5·25-s + 10·27-s − 2·29-s − 8·31-s + 20·33-s − 6·35-s − 20·37-s − 8·39-s + 10·41-s − 43-s + 3·45-s + 47-s + 13·49-s + 6·51-s − 4·53-s + 10·55-s + 6·59-s + ⋯ |
L(s) = 1 | + 1.15·3-s + 0.447·5-s − 2.26·7-s + 9-s + 3.01·11-s − 1.10·13-s + 0.516·15-s + 0.727·17-s − 2.61·21-s − 1.66·23-s + 25-s + 1.92·27-s − 0.371·29-s − 1.43·31-s + 3.48·33-s − 1.01·35-s − 3.28·37-s − 1.28·39-s + 1.56·41-s − 0.152·43-s + 0.447·45-s + 0.145·47-s + 13/7·49-s + 0.840·51-s − 0.549·53-s + 1.34·55-s + 0.781·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2085136 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2085136 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.988329616\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.988329616\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 19 | | \( 1 \) |
good | 3 | $C_2^2$ | \( 1 - 2 T + T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - T - 4 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 4 T + 3 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 8 T + 41 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 2 T - 25 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 10 T + 59 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + T - 42 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - T - 46 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 4 T - 37 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 6 T - 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 12 T + 77 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 2 T - 67 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 9 T + 8 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 8 T - 15 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 12 T + 55 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 8 T - 33 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
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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
−9.760579063697403803129108711091, −9.231347012229685998648068009846, −9.109934664024440756609710152654, −8.578078489088206695794602771094, −8.520097275848638692511335080616, −7.47712127597511159708299247554, −7.22228649559701386862291507096, −6.90598270335006147575555173601, −6.59083925171028664314499355053, −6.17058295471821885490556011945, −5.81077736990966405904312623579, −5.16784165854402464292140616481, −4.45389087770666232967265242029, −3.87460224512887069871937373280, −3.73639881935950836076735897208, −3.14753127930463757655219530761, −2.95591131984208166590982447316, −1.91142896603528396137157560071, −1.72357619854808491325799253027, −0.64101431471754566848328504687,
0.64101431471754566848328504687, 1.72357619854808491325799253027, 1.91142896603528396137157560071, 2.95591131984208166590982447316, 3.14753127930463757655219530761, 3.73639881935950836076735897208, 3.87460224512887069871937373280, 4.45389087770666232967265242029, 5.16784165854402464292140616481, 5.81077736990966405904312623579, 6.17058295471821885490556011945, 6.59083925171028664314499355053, 6.90598270335006147575555173601, 7.22228649559701386862291507096, 7.47712127597511159708299247554, 8.520097275848638692511335080616, 8.578078489088206695794602771094, 9.109934664024440756609710152654, 9.231347012229685998648068009846, 9.760579063697403803129108711091