| L(s) = 1 | − 2·4-s + 8·13-s + 4·16-s + 8·25-s + 4·37-s − 16·52-s + 20·61-s − 8·64-s + 32·73-s − 16·97-s − 16·100-s + 40·109-s − 22·121-s + 127-s + 131-s + 137-s + 139-s − 8·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + 22·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | − 4-s + 2.21·13-s + 16-s + 8/5·25-s + 0.657·37-s − 2.21·52-s + 2.56·61-s − 64-s + 3.74·73-s − 1.62·97-s − 8/5·100-s + 3.83·109-s − 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.657·148-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 + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3111696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3111696 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.437364315\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.437364315\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T^{2} \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 40 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 80 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 56 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 160 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.353836263512789625928165935555, −9.083559597761376916998406488775, −8.604615948543984806954555397492, −8.477249383934390045871409295796, −8.031822174114039517254103200728, −7.75092787742152422470158020981, −6.96859277718540326727871250078, −6.69117887297242671851382966503, −6.34394468186259511321388436319, −5.72250297994792867685532902065, −5.53493084200743304160227091114, −4.93933073014641451114900035031, −4.60221345991105671705217530867, −3.96120476397456095112552159462, −3.67887499882034198321008981467, −3.31426668407820043049485753548, −2.63994859600334458191895790052, −1.90683177118357553204744260386, −1.03503149258991882045679211844, −0.78090895439012213134471147038,
0.78090895439012213134471147038, 1.03503149258991882045679211844, 1.90683177118357553204744260386, 2.63994859600334458191895790052, 3.31426668407820043049485753548, 3.67887499882034198321008981467, 3.96120476397456095112552159462, 4.60221345991105671705217530867, 4.93933073014641451114900035031, 5.53493084200743304160227091114, 5.72250297994792867685532902065, 6.34394468186259511321388436319, 6.69117887297242671851382966503, 6.96859277718540326727871250078, 7.75092787742152422470158020981, 8.031822174114039517254103200728, 8.477249383934390045871409295796, 8.604615948543984806954555397492, 9.083559597761376916998406488775, 9.353836263512789625928165935555
