| L(s) = 1 | − 2·9-s + 12·17-s − 12·41-s − 28·49-s + 4·73-s + 9·81-s − 36·89-s + 40·97-s + 36·113-s + 14·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 24·153-s + 157-s + 163-s + 167-s − 52·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯ |
| L(s) = 1 | − 2/3·9-s + 2.91·17-s − 1.87·41-s − 4·49-s + 0.468·73-s + 81-s − 3.81·89-s + 4.06·97-s + 3.38·113-s + 1.27·121-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 − 1.94·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 4·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.441430735\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.441430735\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 3 | $C_2^3$ | \( 1 + 2 T^{2} - 5 T^{4} + 2 p^{2} T^{6} + p^{4} T^{8} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 11 | $C_2^3$ | \( 1 - 14 T^{2} + 75 T^{4} - 14 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 17 | $C_2^2$ | \( ( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^3$ | \( 1 + 34 T^{2} + 795 T^{4} + 34 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 37 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 + 6 T - 5 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 59 | $C_2^2$ | \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 67 | $C_2^3$ | \( 1 - 62 T^{2} - 645 T^{4} - 62 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 - 2 T - 69 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 83 | $C_2^3$ | \( 1 - 158 T^{2} + 18075 T^{4} - 158 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 + 18 T + 235 T^{2} + 18 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{4} \) |
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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
−5.74246599952465408040931027151, −5.26038273389171924401964893622, −5.20558208093616304434014153866, −5.15589358326795781860947298375, −4.96360028585562418109616791589, −4.82183470449021105462137758311, −4.64938002474910258107931848253, −4.23969484231428526735505651109, −4.09419845765499601783142063264, −3.76884012429317016078092479157, −3.73348785570494535989970991917, −3.37096668253640155189772637869, −3.34762760579528256501330285854, −3.12646890533007671563777803185, −2.89033203408306651148939339900, −2.74089420373019014824015079609, −2.66927592743873606002151491075, −1.97159705551081081513609607887, −1.89592775558909273271737096266, −1.68998137766821502727529725081, −1.52964032107687245472506272896, −1.32350383530765576642765069375, −0.67695095913427092076768563401, −0.54518757079496570592012833740, −0.44778591813116985238565695886,
0.44778591813116985238565695886, 0.54518757079496570592012833740, 0.67695095913427092076768563401, 1.32350383530765576642765069375, 1.52964032107687245472506272896, 1.68998137766821502727529725081, 1.89592775558909273271737096266, 1.97159705551081081513609607887, 2.66927592743873606002151491075, 2.74089420373019014824015079609, 2.89033203408306651148939339900, 3.12646890533007671563777803185, 3.34762760579528256501330285854, 3.37096668253640155189772637869, 3.73348785570494535989970991917, 3.76884012429317016078092479157, 4.09419845765499601783142063264, 4.23969484231428526735505651109, 4.64938002474910258107931848253, 4.82183470449021105462137758311, 4.96360028585562418109616791589, 5.15589358326795781860947298375, 5.20558208093616304434014153866, 5.26038273389171924401964893622, 5.74246599952465408040931027151
