L(s) = 1 | + 2-s + 3-s + 2·4-s + 6-s + 7-s + 5·8-s + 2·12-s + 6·13-s + 14-s + 5·16-s + 2·17-s − 19-s + 21-s + 2·23-s + 5·24-s + 6·26-s − 27-s + 2·28-s − 16·29-s + 8·31-s + 10·32-s + 2·34-s + 7·37-s − 38-s + 6·39-s + 42-s + 16·43-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 4-s + 0.408·6-s + 0.377·7-s + 1.76·8-s + 0.577·12-s + 1.66·13-s + 0.267·14-s + 5/4·16-s + 0.485·17-s − 0.229·19-s + 0.218·21-s + 0.417·23-s + 1.02·24-s + 1.17·26-s − 0.192·27-s + 0.377·28-s − 2.97·29-s + 1.43·31-s + 1.76·32-s + 0.342·34-s + 1.15·37-s − 0.162·38-s + 0.960·39-s + 0.154·42-s + 2.43·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 275625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.898997571\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.898997571\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 3 | $C_2$ | \( 1 - T + T^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - T + p T^{2} \) |
good | 2 | $C_2^2$ | \( 1 - T - T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 2 T - 13 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 2 T - 19 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 8 T + 33 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 7 T + 12 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 10 T + 53 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 14 T + 143 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 10 T + 41 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 7 T - 12 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 11 T + 48 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 7 T - 30 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
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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
−11.12145146344478155155220010411, −10.85566093461833198946429671344, −10.31387055716331359597992124270, −9.858039432859478730609667263865, −9.154941588932228164117478680324, −8.991140583676504853180947362312, −8.088390061054786499721771312548, −8.002953728896567939327330086133, −7.42326257301075120959504043773, −7.20902746717682161342392885518, −6.22736336753547010862975341965, −6.10049495131599517606659189673, −5.63697526841829417306955217803, −4.75283830866069395956696057002, −4.37920712784357924805484873752, −3.91477667406112519572011350224, −3.14320662080991212618876856284, −2.78385899371179767270100688397, −1.61071153691613638870898303207, −1.50776992908578175219466228173,
1.50776992908578175219466228173, 1.61071153691613638870898303207, 2.78385899371179767270100688397, 3.14320662080991212618876856284, 3.91477667406112519572011350224, 4.37920712784357924805484873752, 4.75283830866069395956696057002, 5.63697526841829417306955217803, 6.10049495131599517606659189673, 6.22736336753547010862975341965, 7.20902746717682161342392885518, 7.42326257301075120959504043773, 8.002953728896567939327330086133, 8.088390061054786499721771312548, 8.991140583676504853180947362312, 9.154941588932228164117478680324, 9.858039432859478730609667263865, 10.31387055716331359597992124270, 10.85566093461833198946429671344, 11.12145146344478155155220010411