| L(s) = 1 | − 2·9-s + 16·13-s + 32·37-s − 20·41-s + 28·49-s − 16·53-s − 15·81-s + 60·89-s − 32·117-s + 34·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 108·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | − 2/3·9-s + 4.43·13-s + 5.26·37-s − 3.12·41-s + 4·49-s − 2.19·53-s − 5/3·81-s + 6.35·89-s − 2.95·117-s + 3.09·121-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 + 8.30·169-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 + 0.0708·199-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(9.997701030\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.997701030\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( ( 1 + T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 11 | $C_2^2$ | \( ( 1 - 17 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 17 | $C_2^2$ | \( ( 1 - 25 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 33 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2}( 1 + 10 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
| 41 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 59 | $C_2^2$ | \( ( 1 - 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + 89 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 62 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 65 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + 121 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{4} \) |
| 97 | $C_2^2$ | \( ( 1 - 190 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−6.29635927768635985148295794022, −5.91827650644746452862458490565, −5.64717413589487638144618821148, −5.63878824315742723433952830503, −5.60708497840773924255788381005, −5.20084101836906318870326880345, −4.89061084599680242078097470502, −4.63991035100590968927275818182, −4.39797225976855865700779302141, −4.16606902028033464547514854902, −4.10854851939787514510002284002, −3.98888411143770956685432542982, −3.54377076291573730663509556447, −3.35176816127737789752039240109, −3.17982030399286193943703350669, −3.05078526912244026745410552723, −2.93060223837485628797583161749, −2.33897602197284450798412223476, −2.16036344603606838465693056702, −1.82222806184673331316123481682, −1.76270704469135381107580992722, −1.16816251302182033963238198285, −0.980183265945846575215382953190, −0.73098994878350395815753547228, −0.58130128141877657355539286037,
0.58130128141877657355539286037, 0.73098994878350395815753547228, 0.980183265945846575215382953190, 1.16816251302182033963238198285, 1.76270704469135381107580992722, 1.82222806184673331316123481682, 2.16036344603606838465693056702, 2.33897602197284450798412223476, 2.93060223837485628797583161749, 3.05078526912244026745410552723, 3.17982030399286193943703350669, 3.35176816127737789752039240109, 3.54377076291573730663509556447, 3.98888411143770956685432542982, 4.10854851939787514510002284002, 4.16606902028033464547514854902, 4.39797225976855865700779302141, 4.63991035100590968927275818182, 4.89061084599680242078097470502, 5.20084101836906318870326880345, 5.60708497840773924255788381005, 5.63878824315742723433952830503, 5.64717413589487638144618821148, 5.91827650644746452862458490565, 6.29635927768635985148295794022
