| L(s) = 1 | + 2·3-s − 2·5-s + 4·7-s + 3·9-s − 4·11-s − 4·13-s − 4·15-s − 8·19-s + 8·21-s + 8·23-s + 3·25-s + 4·27-s + 12·29-s + 8·31-s − 8·33-s − 8·35-s + 4·37-s − 8·39-s + 4·41-s − 8·43-s − 6·45-s + 16·47-s + 6·49-s + 4·53-s + 8·55-s − 16·57-s + 12·59-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.894·5-s + 1.51·7-s + 9-s − 1.20·11-s − 1.10·13-s − 1.03·15-s − 1.83·19-s + 1.74·21-s + 1.66·23-s + 3/5·25-s + 0.769·27-s + 2.22·29-s + 1.43·31-s − 1.39·33-s − 1.35·35-s + 0.657·37-s − 1.28·39-s + 0.624·41-s − 1.21·43-s − 0.894·45-s + 2.33·47-s + 6/7·49-s + 0.549·53-s + 1.07·55-s − 2.11·57-s + 1.56·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14745600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14745600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.567856785\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.567856785\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $C_4$ | \( 1 - 4 T + 10 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 8 T + 46 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 - 4 T + 46 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 16 T + 150 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 4 T - 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 12 T + 82 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 110 T^{2} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 4 T + 22 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 - 16 T + 198 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 12 T + 182 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
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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
−8.466281665337689554999685172513, −8.285869651488871260903291141171, −7.998732000583884113064314557984, −7.81750997955453101160926808680, −7.24663625096401282398007234998, −7.08302751521369008680522892424, −6.50263622406099209673595091136, −6.37193305694919498370605758448, −5.29846986803573329753187940041, −5.25752128802517641336915439137, −4.69976905520934099194864110916, −4.62838137197990972694851247939, −4.01900931529327274577309232189, −3.85479650163365616221820829952, −2.89541749342315560378916010563, −2.71734635423501771002740596717, −2.41292000118030871232243765753, −1.98106235214956316241664148490, −1.02733786517929907813512532416, −0.67790247892366330650954601368,
0.67790247892366330650954601368, 1.02733786517929907813512532416, 1.98106235214956316241664148490, 2.41292000118030871232243765753, 2.71734635423501771002740596717, 2.89541749342315560378916010563, 3.85479650163365616221820829952, 4.01900931529327274577309232189, 4.62838137197990972694851247939, 4.69976905520934099194864110916, 5.25752128802517641336915439137, 5.29846986803573329753187940041, 6.37193305694919498370605758448, 6.50263622406099209673595091136, 7.08302751521369008680522892424, 7.24663625096401282398007234998, 7.81750997955453101160926808680, 7.998732000583884113064314557984, 8.285869651488871260903291141171, 8.466281665337689554999685172513
