| L(s) = 1 | − 8·4-s + 40·16-s − 10·25-s + 28·49-s − 160·64-s + 80·100-s − 4·109-s − 14·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 52·169-s + 173-s + 179-s + 181-s + 191-s + 193-s − 224·196-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | − 4·4-s + 10·16-s − 2·25-s + 4·49-s − 20·64-s + 8·100-s − 0.383·109-s − 1.27·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 4·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s − 16·196-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{4} \cdot 29^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{4} \cdot 29^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3650173577\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3650173577\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| good | 2 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 11 | $C_2^2$ | \( ( 1 + 7 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 19 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 - 41 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 - 71 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 53 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 59 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 53 | $C_2^2$ | \( ( 1 + 19 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 61 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 67 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 + T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + 79 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 62 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 + 49 T^{2} + p^{2} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.89969363321014212174241808316, −6.72479145089049766712017799806, −6.16531467087960261605465615692, −6.12100844234200891893254203097, −5.89743988967450134656050730817, −5.65496163549387476363382516234, −5.38568888522453096885479601980, −5.31977653041580870695401398586, −5.22943269081314089635970392005, −4.83225155921523667657761638756, −4.50528740654116124079001807500, −4.37871389256142542574020829343, −4.12907271947267493859959729610, −3.99274686870874226041305619069, −3.93652489556129409685887195005, −3.52605629396906719909437971818, −3.30538821934600939207968873120, −3.06491727752146568228042806351, −2.66394786707359359663027293418, −2.25347376882813811470605930560, −1.82680416816571196489098261312, −1.43833832894140756409834202688, −0.895306534938409028171532575126, −0.76028464924005956211439837042, −0.22269172729142053465133409986,
0.22269172729142053465133409986, 0.76028464924005956211439837042, 0.895306534938409028171532575126, 1.43833832894140756409834202688, 1.82680416816571196489098261312, 2.25347376882813811470605930560, 2.66394786707359359663027293418, 3.06491727752146568228042806351, 3.30538821934600939207968873120, 3.52605629396906719909437971818, 3.93652489556129409685887195005, 3.99274686870874226041305619069, 4.12907271947267493859959729610, 4.37871389256142542574020829343, 4.50528740654116124079001807500, 4.83225155921523667657761638756, 5.22943269081314089635970392005, 5.31977653041580870695401398586, 5.38568888522453096885479601980, 5.65496163549387476363382516234, 5.89743988967450134656050730817, 6.12100844234200891893254203097, 6.16531467087960261605465615692, 6.72479145089049766712017799806, 6.89969363321014212174241808316
