| L(s) = 1 | + 40·13-s + 64·25-s − 256·37-s + 124·49-s − 384·73-s + 256·97-s + 472·109-s − 156·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 324·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯ |
| L(s) = 1 | + 3.07·13-s + 2.55·25-s − 6.91·37-s + 2.53·49-s − 5.26·73-s + 2.63·97-s + 4.33·109-s − 1.28·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 1.91·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + 0.00473·211-s + 0.00448·223-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.888124965\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.888124965\) |
| \(L(2)\) |
|
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 | $C_2^2$ | \( ( 1 - 32 T^{2} + p^{4} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 - 62 T^{2} + p^{4} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 78 T^{2} + p^{4} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 10 T + p^{2} T^{2} )^{4} \) |
| 17 | $C_2^2$ | \( ( 1 + p^{4} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 82 T^{2} + p^{4} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 738 T^{2} + p^{4} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 1440 T^{2} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 1762 T^{2} + p^{4} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 64 T + p^{2} T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 - 3200 T^{2} + p^{4} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + 1138 T^{2} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 4098 T^{2} + p^{4} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 5280 T^{2} + p^{4} T^{4} )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 61 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 + 3218 T^{2} + p^{4} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 5598 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 96 T + p^{2} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 + 8482 T^{2} + p^{4} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 1902 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 12800 T^{2} + p^{4} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 64 T + p^{2} T^{2} )^{4} \) |
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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
−6.09528831896436137442643889742, −6.08588150611455387424130329152, −6.03464821197049460312409008164, −5.49726218842155526413646282000, −5.36330129971576921518128053138, −5.20548980439383507614311894392, −4.97345281264302279251306664977, −4.72456320008307936616678980963, −4.59331773796925220861342709514, −4.37267645516667706810350332332, −3.79170741273606115522921178278, −3.71798547277911636652297423690, −3.56876396728138768959341079755, −3.45817323666996533657950087036, −3.43390380016485527899823248234, −2.79435125971884525132606160618, −2.67914055034258407827646936967, −2.53559789120233480042232808236, −1.90100910591100051825191007665, −1.70613043089648358600698056359, −1.43405237677503572367803161961, −1.38711616462805954730759411536, −1.01294536282631318344009251211, −0.55868687081613015739409013733, −0.24547166614821798656564145679,
0.24547166614821798656564145679, 0.55868687081613015739409013733, 1.01294536282631318344009251211, 1.38711616462805954730759411536, 1.43405237677503572367803161961, 1.70613043089648358600698056359, 1.90100910591100051825191007665, 2.53559789120233480042232808236, 2.67914055034258407827646936967, 2.79435125971884525132606160618, 3.43390380016485527899823248234, 3.45817323666996533657950087036, 3.56876396728138768959341079755, 3.71798547277911636652297423690, 3.79170741273606115522921178278, 4.37267645516667706810350332332, 4.59331773796925220861342709514, 4.72456320008307936616678980963, 4.97345281264302279251306664977, 5.20548980439383507614311894392, 5.36330129971576921518128053138, 5.49726218842155526413646282000, 6.03464821197049460312409008164, 6.08588150611455387424130329152, 6.09528831896436137442643889742
