| L(s) = 1 | + 3·9-s − 6·11-s + 2·19-s − 10·29-s − 16·31-s − 14·41-s − 5·49-s + 10·59-s + 10·61-s + 6·71-s + 32·79-s + 9·81-s + 14·89-s − 18·99-s + 18·101-s + 56·109-s + 31·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 22·169-s + ⋯ |
| L(s) = 1 | + 9-s − 1.80·11-s + 0.458·19-s − 1.85·29-s − 2.87·31-s − 2.18·41-s − 5/7·49-s + 1.30·59-s + 1.28·61-s + 0.712·71-s + 3.60·79-s + 81-s + 1.48·89-s − 1.80·99-s + 1.79·101-s + 5.36·109-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 + 1.69·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.737131206\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.737131206\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| good | 3 | $C_2$$\times$$C_2^2$ | \( ( 1 - p T^{2} )^{2}( 1 + p T^{2} + p^{2} T^{4} ) \) |
| 7 | $C_2^3$ | \( 1 + 5 T^{2} - 24 T^{4} + 5 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^3$ | \( 1 - 15 T^{2} - 64 T^{4} - 15 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2} \) |
| 23 | $C_2^3$ | \( 1 - 3 T^{2} - 520 T^{4} - 3 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 + 5 T - 4 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 37 | $C_2^3$ | \( 1 + 65 T^{2} + 2856 T^{4} + 65 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_2^2$ | \( ( 1 + 7 T + 8 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^3$ | \( 1 + 5 T^{2} - 1824 T^{4} + 5 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $C_2^2$ | \( ( 1 - 30 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 5 T - 34 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 5 T - 36 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^3$ | \( 1 - 35 T^{2} - 3264 T^{4} - 35 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 - 3 T - 62 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 7 T - 40 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^3$ | \( 1 + 73 T^{2} - 4080 T^{4} + 73 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−7.01543148840224216825071019070, −6.76546306165468248768041626248, −6.38126906711376896659482517145, −6.37098438775061410126775251949, −5.86826651645782380958003358713, −5.80498441206477212824978277536, −5.40171219128121160443062457696, −5.38437232919076095023508226281, −5.20768396828488599641568330479, −4.82997563268097608512859044859, −4.77267191781205223333612541309, −4.53339359385173684778176438070, −4.13405823844536446109174462772, −3.72000397069170886026498023035, −3.60297079475479156408754319653, −3.43849065094132470166468523251, −3.23359060522149954992502521669, −3.07715631262917029914693650383, −2.19566575403911816895561774177, −2.09626907368375395436704493030, −2.03174172627228209767222686356, −2.01070486842871243403993685576, −1.23707753508114821971067660253, −0.74415245683020832165539587452, −0.32729642532591563108742734981,
0.32729642532591563108742734981, 0.74415245683020832165539587452, 1.23707753508114821971067660253, 2.01070486842871243403993685576, 2.03174172627228209767222686356, 2.09626907368375395436704493030, 2.19566575403911816895561774177, 3.07715631262917029914693650383, 3.23359060522149954992502521669, 3.43849065094132470166468523251, 3.60297079475479156408754319653, 3.72000397069170886026498023035, 4.13405823844536446109174462772, 4.53339359385173684778176438070, 4.77267191781205223333612541309, 4.82997563268097608512859044859, 5.20768396828488599641568330479, 5.38437232919076095023508226281, 5.40171219128121160443062457696, 5.80498441206477212824978277536, 5.86826651645782380958003358713, 6.37098438775061410126775251949, 6.38126906711376896659482517145, 6.76546306165468248768041626248, 7.01543148840224216825071019070
