| L(s) = 1 | + 16·23-s − 16·25-s + 48·47-s + 12·49-s + 16·71-s + 32·73-s + 12·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 44·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯ |
| L(s) = 1 | + 3.33·23-s − 3.19·25-s + 7.00·47-s + 12/7·49-s + 1.89·71-s + 3.74·73-s + 1.09·121-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 + 3.38·169-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 + 0.0688·211-s + 0.0669·223-s + 0.0663·227-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.584268684\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.584268684\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( ( 1 + 8 T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 32 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 29 | $C_2^2$ | \( ( 1 + 8 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 64 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 42 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{4} \) |
| 53 | $C_2^2$ | \( ( 1 - 56 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 61 | $C_2^2$ | \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + 102 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 73 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - 150 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 64 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
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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
−7.05179056661844512749662949148, −6.70982732685936686889925617239, −6.66862840200827491969427911818, −6.38345434845708658101467688378, −6.13685179323515020618591546317, −5.73557438393853998004627628029, −5.54879725037018981711901761452, −5.47145482224507658072708235653, −5.43740440800095212987232032961, −5.01846321955504386714905129998, −4.78903020089471820399195516706, −4.29821067243884985886996079293, −4.21136281533773992429205884677, −3.96013668216392425451004654620, −3.89360831860130700146941109336, −3.40199132105841922762692964135, −3.34715330281123951424148908266, −2.80770860167350525144284991224, −2.48149297543827828716775791813, −2.42299520525990374527095817847, −2.06509222166753356777367732558, −1.82779746240293080519351279816, −0.963600920384473100458553095166, −0.919482224341953366354628498840, −0.61226285962704655126300726828,
0.61226285962704655126300726828, 0.919482224341953366354628498840, 0.963600920384473100458553095166, 1.82779746240293080519351279816, 2.06509222166753356777367732558, 2.42299520525990374527095817847, 2.48149297543827828716775791813, 2.80770860167350525144284991224, 3.34715330281123951424148908266, 3.40199132105841922762692964135, 3.89360831860130700146941109336, 3.96013668216392425451004654620, 4.21136281533773992429205884677, 4.29821067243884985886996079293, 4.78903020089471820399195516706, 5.01846321955504386714905129998, 5.43740440800095212987232032961, 5.47145482224507658072708235653, 5.54879725037018981711901761452, 5.73557438393853998004627628029, 6.13685179323515020618591546317, 6.38345434845708658101467688378, 6.66862840200827491969427911818, 6.70982732685936686889925617239, 7.05179056661844512749662949148
