| L(s) = 1 | + 2-s − 4-s − 4·5-s − 3·8-s − 4·9-s − 4·10-s − 8·11-s − 16-s + 5·17-s − 4·18-s − 6·19-s + 4·20-s − 8·22-s − 6·23-s + 2·25-s + 5·32-s + 5·34-s + 4·36-s − 6·37-s − 6·38-s + 12·40-s + 8·44-s + 16·45-s − 6·46-s − 4·49-s + 2·50-s + 32·55-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 1.78·5-s − 1.06·8-s − 4/3·9-s − 1.26·10-s − 2.41·11-s − 1/4·16-s + 1.21·17-s − 0.942·18-s − 1.37·19-s + 0.894·20-s − 1.70·22-s − 1.25·23-s + 2/5·25-s + 0.883·32-s + 0.857·34-s + 2/3·36-s − 0.986·37-s − 0.973·38-s + 1.89·40-s + 1.20·44-s + 2.38·45-s − 0.884·46-s − 4/7·49-s + 0.282·50-s + 4.31·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 183872 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 183872 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + p T^{2} \) |
| 13 | $C_2$ | \( 1 + p T^{2} \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 6 T + p T^{2} ) \) |
| good | 3 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 28 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 48 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 - 126 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 122 T^{2} + p^{2} T^{4} \) |
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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
−8.426484696513399994730187699646, −8.185173332450664765302448544174, −7.74872897501832974624995215110, −7.60746011239254894703380630880, −6.62333731764880742262118022713, −5.93770959874546765867746339172, −5.59006132551563029107011912337, −5.12152287440228841362856662894, −4.57219388638396145336864251083, −3.92848048627018603689395177954, −3.49649181722721654388286813753, −2.92820439332086546686214899541, −2.29419708627815123330071108267, 0, 0,
2.29419708627815123330071108267, 2.92820439332086546686214899541, 3.49649181722721654388286813753, 3.92848048627018603689395177954, 4.57219388638396145336864251083, 5.12152287440228841362856662894, 5.59006132551563029107011912337, 5.93770959874546765867746339172, 6.62333731764880742262118022713, 7.60746011239254894703380630880, 7.74872897501832974624995215110, 8.185173332450664765302448544174, 8.426484696513399994730187699646
