| L(s) = 1 | − 8·4-s − 12·5-s + 48·16-s + 24·19-s + 96·20-s + 72·25-s − 98·49-s − 216·59-s + 24·61-s − 256·64-s − 192·76-s − 576·80-s + 62·81-s − 288·95-s − 576·100-s − 456·101-s + 484·121-s − 300·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | − 2·4-s − 2.39·5-s + 3·16-s + 1.26·19-s + 24/5·20-s + 2.87·25-s − 2·49-s − 3.66·59-s + 0.393·61-s − 4·64-s − 2.52·76-s − 7.19·80-s + 0.765·81-s − 3.03·95-s − 5.75·100-s − 4.51·101-s + 4·121-s − 2.39·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.3182444831\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3182444831\) |
| \(L(2)\) |
|
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 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 + 12 T + 72 T^{2} + 12 p^{2} T^{3} + p^{4} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| good | 3 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 4 T + 8 T^{2} - 4 p^{2} T^{3} + p^{4} T^{4} )( 1 + 4 T + 8 T^{2} + 4 p^{2} T^{3} + p^{4} T^{4} ) \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 13 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 36 T + 648 T^{2} - 36 p^{2} T^{3} + p^{4} T^{4} )( 1 + 36 T + 648 T^{2} + 36 p^{2} T^{3} + p^{4} T^{4} ) \) |
| 17 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 - 12 T + 72 T^{2} - 12 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 10 T + p^{2} T^{2} )^{2}( 1 + 10 T + p^{2} T^{2} )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 37 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 43 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 47 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 53 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 59 | $C_2^2$ | \( ( 1 + 108 T + 5832 T^{2} + 108 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 12 T + 72 T^{2} - 12 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 71 | $C_2^2$ | \( ( 1 + 2018 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 + 4418 T^{2} + p^{4} T^{4} )^{2} \) |
| 83 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 36 T + 648 T^{2} - 36 p^{2} T^{3} + p^{4} T^{4} )( 1 + 36 T + 648 T^{2} + 36 p^{2} T^{3} + p^{4} T^{4} ) \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 97 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.286107074219089479395741965990, −7.955776684920225992121566234330, −7.911928058486773544664066857232, −7.80303192142351709361792158239, −7.79982594964652747047628304590, −7.03402991580536242203845063103, −6.93519436537513136961951526737, −6.80813935986033871926019999644, −6.25281723231354244621359890066, −5.84458822010063402861695961571, −5.63237777368329322525500177538, −5.35618545705637081861737893502, −5.01415844445593604116716937024, −4.65981357010486468160347394402, −4.44868395584395230043331150407, −4.28337236439512769285798333418, −4.06233371159878848066019450397, −3.44956768307141996389491915632, −3.36543292297110866241218375783, −3.15139777802029909916556772511, −2.82414965201592099029278439409, −1.78617178049995648315934861482, −1.33639873149900986017596471455, −0.69468955903963788946698604723, −0.24121789219011210345184985593,
0.24121789219011210345184985593, 0.69468955903963788946698604723, 1.33639873149900986017596471455, 1.78617178049995648315934861482, 2.82414965201592099029278439409, 3.15139777802029909916556772511, 3.36543292297110866241218375783, 3.44956768307141996389491915632, 4.06233371159878848066019450397, 4.28337236439512769285798333418, 4.44868395584395230043331150407, 4.65981357010486468160347394402, 5.01415844445593604116716937024, 5.35618545705637081861737893502, 5.63237777368329322525500177538, 5.84458822010063402861695961571, 6.25281723231354244621359890066, 6.80813935986033871926019999644, 6.93519436537513136961951526737, 7.03402991580536242203845063103, 7.79982594964652747047628304590, 7.80303192142351709361792158239, 7.911928058486773544664066857232, 7.955776684920225992121566234330, 8.286107074219089479395741965990
