| L(s) = 1 | + 4-s − 2·9-s − 6·11-s − 2·19-s − 24·29-s − 16·31-s − 2·36-s + 12·41-s − 6·44-s − 2·49-s − 16·61-s − 64-s − 2·76-s − 20·79-s + 9·81-s + 12·89-s + 12·99-s − 24·101-s − 32·109-s − 24·116-s + 31·121-s − 16·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 2/3·9-s − 1.80·11-s − 0.458·19-s − 4.45·29-s − 2.87·31-s − 1/3·36-s + 1.87·41-s − 0.904·44-s − 2/7·49-s − 2.04·61-s − 1/8·64-s − 0.229·76-s − 2.25·79-s + 81-s + 1.27·89-s + 1.20·99-s − 2.38·101-s − 3.06·109-s − 2.22·116-s + 2.81·121-s − 1.43·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2001141094\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2001141094\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 5 | | \( 1 \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| good | 3 | $C_2^3$ | \( 1 + 2 T^{2} - 5 T^{4} + 2 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 25 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^3$ | \( 1 - 2 T^{2} - 285 T^{4} - 2 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \) |
| 23 | $C_2^3$ | \( 1 - 35 T^{2} + 696 T^{4} - 35 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 + 8 T + 33 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^3$ | \( 1 + 25 T^{2} - 744 T^{4} + 25 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 + 13 T^{2} - 2040 T^{4} + 13 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^3$ | \( 1 + 25 T^{2} - 2184 T^{4} + 25 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 8 T + 3 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^3$ | \( 1 + 70 T^{2} + 411 T^{4} + 70 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 73 | $C_2^3$ | \( 1 + 130 T^{2} + 11571 T^{4} + 130 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 + 10 T + 21 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 89 | $C_2^2$ | \( ( 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 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
−8.378784040372854806813467089956, −7.87350964036481138544346641853, −7.69474732751658795411804675735, −7.69140614198270563361395868114, −7.48935748515932652930798818221, −7.15540519822383930588653210999, −6.88967444215785381371346319154, −6.65687020368110012562934994432, −6.15333425103588563995511785131, −5.92201105205790033241447731572, −5.59757163456965982266355245750, −5.57281077153239740947632813509, −5.35327274963223102197616660979, −5.15610651410940936830699921442, −4.48386521550546793761825348765, −4.33846452465567365361262313465, −3.91466826864249975178918350419, −3.70857199231072531366990993898, −3.20741109548216959135108255104, −3.05920399620923423330145031730, −2.55997119724191358317682149515, −2.19839207267687866583269613881, −1.85556441215371037254042413834, −1.56897112805393215768802291785, −0.16874270616552545056516906016,
0.16874270616552545056516906016, 1.56897112805393215768802291785, 1.85556441215371037254042413834, 2.19839207267687866583269613881, 2.55997119724191358317682149515, 3.05920399620923423330145031730, 3.20741109548216959135108255104, 3.70857199231072531366990993898, 3.91466826864249975178918350419, 4.33846452465567365361262313465, 4.48386521550546793761825348765, 5.15610651410940936830699921442, 5.35327274963223102197616660979, 5.57281077153239740947632813509, 5.59757163456965982266355245750, 5.92201105205790033241447731572, 6.15333425103588563995511785131, 6.65687020368110012562934994432, 6.88967444215785381371346319154, 7.15540519822383930588653210999, 7.48935748515932652930798818221, 7.69140614198270563361395868114, 7.69474732751658795411804675735, 7.87350964036481138544346641853, 8.378784040372854806813467089956
