| L(s) = 1 | − 4·4-s + 12·16-s − 12·25-s − 48·37-s − 32·64-s + 48·100-s + 24·109-s − 44·121-s + 127-s + 131-s + 137-s + 139-s + 192·148-s + 149-s + 151-s + 157-s + 163-s + 167-s − 48·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | − 2·4-s + 3·16-s − 2.39·25-s − 7.89·37-s − 4·64-s + 24/5·100-s + 2.29·109-s − 4·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 15.7·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 3.69·169-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3742317468\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3742317468\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2}( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 13 | $C_2^2$ | \( ( 1 + 24 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2}( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 29 | $C_2^2$ | \( ( 1 + 40 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 37 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{4} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 53 | $C_2^2$ | \( ( 1 + 56 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 61 | $C_2^2$ | \( ( 1 + 120 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 - 96 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 89 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2}( 1 + 16 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 + 144 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−6.75943423658900417910456068094, −6.28756890209265341045677933621, −6.06150434051900856513791630418, −6.02969425669910435466790852438, −5.60365093259543978271686260448, −5.49310898582601400175030855731, −5.21416885316936654691699974897, −5.09397726759886403979786877219, −5.05334656711871377339260849819, −4.69405975585808992869771924693, −4.50478577963547302249280201088, −4.16989267590761182429430584256, −3.75606377384864649703008184280, −3.74414134670296730798317355460, −3.58932959272581837705919122845, −3.44900764078740645111408528794, −3.25940535669326296383125506206, −2.61979218123091081845200045651, −2.53770071686067486774078085380, −1.87072111657475047936625685676, −1.83988697303736087905377634060, −1.42526103099842909145567247411, −1.35019075044378241658697112143, −0.40285655405865065036678531618, −0.23133110380301357356979772005,
0.23133110380301357356979772005, 0.40285655405865065036678531618, 1.35019075044378241658697112143, 1.42526103099842909145567247411, 1.83988697303736087905377634060, 1.87072111657475047936625685676, 2.53770071686067486774078085380, 2.61979218123091081845200045651, 3.25940535669326296383125506206, 3.44900764078740645111408528794, 3.58932959272581837705919122845, 3.74414134670296730798317355460, 3.75606377384864649703008184280, 4.16989267590761182429430584256, 4.50478577963547302249280201088, 4.69405975585808992869771924693, 5.05334656711871377339260849819, 5.09397726759886403979786877219, 5.21416885316936654691699974897, 5.49310898582601400175030855731, 5.60365093259543978271686260448, 6.02969425669910435466790852438, 6.06150434051900856513791630418, 6.28756890209265341045677933621, 6.75943423658900417910456068094
