| L(s) = 1 | − 8·4-s + 9-s + 40·16-s − 10·25-s − 8·36-s + 14·49-s − 160·64-s − 4·79-s − 8·81-s + 80·100-s + 44·109-s − 26·121-s + 127-s + 131-s + 137-s + 139-s + 40·144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 38·169-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
| L(s) = 1 | − 4·4-s + 1/3·9-s + 10·16-s − 2·25-s − 4/3·36-s + 2·49-s − 20·64-s − 0.450·79-s − 8/9·81-s + 8·100-s + 4.21·109-s − 2.36·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 10/3·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 2.92·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3081375071\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3081375071\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| good | 2 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2}( 1 + 3 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + 19 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 29 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 29 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2}( 1 + 9 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 37 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 43 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 47 | $C_2^2$ | \( ( 1 + 31 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 61 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 67 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2}( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + T + p T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 - 86 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 97 | $C_2^2$ | \( ( 1 - 149 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−10.04464913442371129729613491573, −9.897870467817826874600174401912, −9.579282568493256666071670355384, −9.224795699877389279750112784389, −9.109313852922315099750217969886, −8.870322178775433819924699679718, −8.489223512008288469056209217900, −8.236810291145322659413230385922, −8.123299798078636550615703189610, −7.66739092545290842768189535246, −7.30038568161409091706010431392, −7.26360276363798949662648662762, −6.21577155880049371115133005822, −6.15362876635385216636892125316, −5.62383035160571644093125849253, −5.45555262408562740704452774358, −5.15802979285512709996728673531, −4.71280274439209525804059574563, −4.28165375687532667794677697703, −4.21850871250961815429634595116, −3.82719619637920896474875253525, −3.47110627331364208458537828722, −2.94173244415565854990122695373, −1.72878415582342817772241598425, −0.66368167937209263439652868992,
0.66368167937209263439652868992, 1.72878415582342817772241598425, 2.94173244415565854990122695373, 3.47110627331364208458537828722, 3.82719619637920896474875253525, 4.21850871250961815429634595116, 4.28165375687532667794677697703, 4.71280274439209525804059574563, 5.15802979285512709996728673531, 5.45555262408562740704452774358, 5.62383035160571644093125849253, 6.15362876635385216636892125316, 6.21577155880049371115133005822, 7.26360276363798949662648662762, 7.30038568161409091706010431392, 7.66739092545290842768189535246, 8.123299798078636550615703189610, 8.236810291145322659413230385922, 8.489223512008288469056209217900, 8.870322178775433819924699679718, 9.109313852922315099750217969886, 9.224795699877389279750112784389, 9.579282568493256666071670355384, 9.897870467817826874600174401912, 10.04464913442371129729613491573
