| L(s) = 1 | − 5·9-s + 6·11-s − 2·19-s + 24·29-s + 14·31-s + 24·41-s − 2·49-s + 18·59-s + 2·61-s − 26·79-s + 9·81-s + 30·89-s − 30·99-s − 30·101-s − 2·109-s + 31·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 44·169-s + 10·171-s + ⋯ |
| L(s) = 1 | − 5/3·9-s + 1.80·11-s − 0.458·19-s + 4.45·29-s + 2.51·31-s + 3.74·41-s − 2/7·49-s + 2.34·59-s + 0.256·61-s − 2.92·79-s + 81-s + 3.17·89-s − 3.01·99-s − 2.98·101-s − 0.191·109-s + 2.81·121-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 + 3.38·169-s + 0.764·171-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.616695058\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.616695058\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| good | 3 | $C_2^3$ | \( 1 + 5 T^{2} + 16 T^{4} + 5 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^3$ | \( 1 + 25 T^{2} + 336 T^{4} + 25 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \) |
| 23 | $C_2^3$ | \( 1 + 37 T^{2} + 840 T^{4} + 37 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$$\times$$C_2^2$ | \( ( 1 + 26 T^{2} + p^{2} T^{4} )( 1 + 47 T^{2} + p^{2} T^{4} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 + 13 T^{2} - 2040 T^{4} + 13 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^3$ | \( 1 + 97 T^{2} + 6600 T^{4} + 97 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 - 9 T + 22 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2}( 1 + 13 T + p T^{2} )^{2} \) |
| 67 | $C_2^3$ | \( 1 + 85 T^{2} + 2736 T^{4} + 85 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 73 | $C_2^3$ | \( 1 + 145 T^{2} + 15696 T^{4} + 145 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 17 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 15 T + 136 T^{2} - 15 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 94 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−7.78383071455577516978756509009, −7.11819984222519069098002794999, −7.03205833141030878966178964370, −6.61471578461042561160903581751, −6.59397572795414659931038445589, −6.38719467063778965646577524395, −6.00337271504048701634095107857, −6.00075083894100920395407015869, −5.92711842536935524451389851746, −5.21655766996772904948754582958, −5.10325235297456487298999083648, −4.94392405431303223645198880202, −4.50534004762689415947944390747, −4.29848960580906198104923715898, −4.11086253759040976385742962966, −3.86496266686828701105150565772, −3.58943943013354949596307518958, −2.83416821354201037765539083348, −2.78977830424068671079950438877, −2.74432547150436190087870837135, −2.54908013316194272179441710969, −1.89529636746624568587643586417, −1.20192676082104167361428121210, −0.995514160080586106266598406927, −0.67332281596585373974923149882,
0.67332281596585373974923149882, 0.995514160080586106266598406927, 1.20192676082104167361428121210, 1.89529636746624568587643586417, 2.54908013316194272179441710969, 2.74432547150436190087870837135, 2.78977830424068671079950438877, 2.83416821354201037765539083348, 3.58943943013354949596307518958, 3.86496266686828701105150565772, 4.11086253759040976385742962966, 4.29848960580906198104923715898, 4.50534004762689415947944390747, 4.94392405431303223645198880202, 5.10325235297456487298999083648, 5.21655766996772904948754582958, 5.92711842536935524451389851746, 6.00075083894100920395407015869, 6.00337271504048701634095107857, 6.38719467063778965646577524395, 6.59397572795414659931038445589, 6.61471578461042561160903581751, 7.03205833141030878966178964370, 7.11819984222519069098002794999, 7.78383071455577516978756509009
