| L(s) = 1 | + 4·4-s + 4·16-s − 8·25-s − 16·37-s − 28·43-s − 16·64-s + 8·67-s − 16·79-s − 32·100-s + 68·109-s − 20·121-s + 127-s + 131-s + 137-s + 139-s − 64·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + 28·169-s − 112·172-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
| L(s) = 1 | + 2·4-s + 16-s − 8/5·25-s − 2.63·37-s − 4.26·43-s − 2·64-s + 0.977·67-s − 1.80·79-s − 3.19·100-s + 6.51·109-s − 1.81·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 5.26·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2.15·169-s − 8.53·172-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.441726221\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.441726221\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 5 | $C_2^2$ | \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 28 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 35 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 56 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 35 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 + 76 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{4} \) |
| 47 | $C_2^2$ | \( ( 1 + 40 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 104 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 98 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 95 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 71 | $C_2^2$ | \( ( 1 - 134 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 172 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 119 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−6.83319493251373109573172059498, −6.73160133576794666526167745560, −6.71452128124924412308042354250, −6.30059493774842912022288552537, −5.90061519650483049965513452327, −5.81106970216465401383198068022, −5.79640001796900568062568531768, −5.34762215962010658576289169963, −5.09697202808740114855547135786, −5.01825711218003543472877545258, −4.52564831901881287593505763555, −4.41702198243998936206426881347, −4.27648783720070339870586111494, −3.66054628467012157156542461863, −3.58292376666547518646566400215, −3.22793677079083761948203997340, −3.14775754718984525975416580052, −2.97158607981496238445906936250, −2.46993964789137036973202544275, −1.97323895603070491724512981918, −1.94569743290888129235309167133, −1.83388915671197582229556858611, −1.62207956988385369241043138511, −0.915507227025875453357793723236, −0.21301761024486320623067274634,
0.21301761024486320623067274634, 0.915507227025875453357793723236, 1.62207956988385369241043138511, 1.83388915671197582229556858611, 1.94569743290888129235309167133, 1.97323895603070491724512981918, 2.46993964789137036973202544275, 2.97158607981496238445906936250, 3.14775754718984525975416580052, 3.22793677079083761948203997340, 3.58292376666547518646566400215, 3.66054628467012157156542461863, 4.27648783720070339870586111494, 4.41702198243998936206426881347, 4.52564831901881287593505763555, 5.01825711218003543472877545258, 5.09697202808740114855547135786, 5.34762215962010658576289169963, 5.79640001796900568062568531768, 5.81106970216465401383198068022, 5.90061519650483049965513452327, 6.30059493774842912022288552537, 6.71452128124924412308042354250, 6.73160133576794666526167745560, 6.83319493251373109573172059498
