| L(s) = 1 | − 2·3-s − 2·5-s + 2·7-s + 2·9-s + 2·13-s + 4·15-s + 2·17-s + 8·19-s − 4·21-s − 10·23-s − 25-s − 6·27-s − 4·35-s + 10·37-s − 4·39-s + 12·41-s + 6·43-s − 4·45-s − 14·47-s + 2·49-s − 4·51-s + 2·53-s − 16·57-s + 8·59-s − 4·61-s + 4·63-s − 4·65-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.894·5-s + 0.755·7-s + 2/3·9-s + 0.554·13-s + 1.03·15-s + 0.485·17-s + 1.83·19-s − 0.872·21-s − 2.08·23-s − 1/5·25-s − 1.15·27-s − 0.676·35-s + 1.64·37-s − 0.640·39-s + 1.87·41-s + 0.914·43-s − 0.596·45-s − 2.04·47-s + 2/7·49-s − 0.560·51-s + 0.274·53-s − 2.11·57-s + 1.04·59-s − 0.512·61-s + 0.503·63-s − 0.496·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 102400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 102400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8854479568\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8854479568\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 18 T + 162 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
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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
−11.90610077894629688601717844975, −11.37640160458337348389505031560, −11.03811215650973180610571614677, −10.84688550840180266457463482690, −9.890225631806539205448181678487, −9.704867874929865178424189540651, −9.263756994117660218074196810189, −8.232194008465131626483432211018, −8.005957393011283377556188545346, −7.64506178013416516018906516900, −7.21272644162031528486031263166, −6.29867084661856279217229405749, −5.93547372723224016811702573831, −5.50630495973484261816459782947, −4.92567796347191702071413721364, −4.08707047708737683569632907038, −3.95375881348507857996110734004, −2.96026386145765863367697844051, −1.81572903787505839753233493118, −0.76743892918537886287415781356,
0.76743892918537886287415781356, 1.81572903787505839753233493118, 2.96026386145765863367697844051, 3.95375881348507857996110734004, 4.08707047708737683569632907038, 4.92567796347191702071413721364, 5.50630495973484261816459782947, 5.93547372723224016811702573831, 6.29867084661856279217229405749, 7.21272644162031528486031263166, 7.64506178013416516018906516900, 8.005957393011283377556188545346, 8.232194008465131626483432211018, 9.263756994117660218074196810189, 9.704867874929865178424189540651, 9.890225631806539205448181678487, 10.84688550840180266457463482690, 11.03811215650973180610571614677, 11.37640160458337348389505031560, 11.90610077894629688601717844975
