| L(s) = 1 | + 4·5-s + 9-s + 8·17-s + 11·25-s − 4·29-s − 8·37-s + 12·41-s + 4·45-s − 6·49-s + 4·53-s − 12·61-s − 16·73-s + 81-s + 32·85-s + 12·89-s + 16·97-s + 12·101-s − 12·109-s + 6·121-s + 24·125-s + 127-s + 131-s + 137-s + 139-s − 16·145-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 1.78·5-s + 1/3·9-s + 1.94·17-s + 11/5·25-s − 0.742·29-s − 1.31·37-s + 1.87·41-s + 0.596·45-s − 6/7·49-s + 0.549·53-s − 1.53·61-s − 1.87·73-s + 1/9·81-s + 3.47·85-s + 1.27·89-s + 1.62·97-s + 1.19·101-s − 1.14·109-s + 6/11·121-s + 2.14·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1.32·145-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 460800 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460800 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.102555772\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.102555772\) |
| \(L(\frac{3}{2})\) |
|
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$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
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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
−8.715051704485126753869246644855, −8.066132725591169059383712780439, −7.57465064060797654237372569489, −7.25843391351741208963448809078, −6.64479440226593691027462612408, −6.12443286941683078981500888806, −5.68195896020875081111167935487, −5.53458000608304393865103922563, −4.83929020295476241899652709575, −4.36463658504323351341021473431, −3.41639482527061604903675336016, −3.12661147993530431750441461946, −2.27003827651712229587169600316, −1.70211655868142993720698982735, −1.03876412690559493571282883382,
1.03876412690559493571282883382, 1.70211655868142993720698982735, 2.27003827651712229587169600316, 3.12661147993530431750441461946, 3.41639482527061604903675336016, 4.36463658504323351341021473431, 4.83929020295476241899652709575, 5.53458000608304393865103922563, 5.68195896020875081111167935487, 6.12443286941683078981500888806, 6.64479440226593691027462612408, 7.25843391351741208963448809078, 7.57465064060797654237372569489, 8.066132725591169059383712780439, 8.715051704485126753869246644855
