| L(s) = 1 | − 3-s − 4-s − 2·9-s + 12-s + 16-s + 11·17-s + 6·25-s + 5·27-s − 9·29-s + 2·36-s − 43-s − 48-s − 49-s − 11·51-s − 5·53-s + 13·61-s − 64-s − 11·68-s − 6·75-s − 15·79-s + 81-s + 9·87-s − 6·100-s + 11·101-s − 103-s + 26·107-s − 5·108-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1/2·4-s − 2/3·9-s + 0.288·12-s + 1/4·16-s + 2.66·17-s + 6/5·25-s + 0.962·27-s − 1.67·29-s + 1/3·36-s − 0.152·43-s − 0.144·48-s − 1/7·49-s − 1.54·51-s − 0.686·53-s + 1.66·61-s − 1/8·64-s − 1.33·68-s − 0.692·75-s − 1.68·79-s + 1/9·81-s + 0.964·87-s − 3/5·100-s + 1.09·101-s − 0.0985·103-s + 2.51·107-s − 0.481·108-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.147768520\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.147768520\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_2$ | \( 1 + T + p T^{2} \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| 13 | $C_2$ | \( 1 + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 3 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 53 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 47 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 115 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 5 T^{2} + p^{2} T^{4} \) |
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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
−8.801865297960571300465901701440, −8.355259144899635669374028917046, −7.977495753198467776695010790032, −7.34421753504590671509703779979, −7.13950789800159507342557907464, −6.23289191305093910306445489370, −5.94025866236808335791283345861, −5.31115087019525329111918769984, −5.22265996614244128825894790351, −4.50681747145871408585541457793, −3.65705823831213042471955558700, −3.34349869514538042531219722257, −2.67985472529534111917542756516, −1.55844681266113046809237210397, −0.69212136137643190815272938222,
0.69212136137643190815272938222, 1.55844681266113046809237210397, 2.67985472529534111917542756516, 3.34349869514538042531219722257, 3.65705823831213042471955558700, 4.50681747145871408585541457793, 5.22265996614244128825894790351, 5.31115087019525329111918769984, 5.94025866236808335791283345861, 6.23289191305093910306445489370, 7.13950789800159507342557907464, 7.34421753504590671509703779979, 7.977495753198467776695010790032, 8.355259144899635669374028917046, 8.801865297960571300465901701440
