| L(s) = 1 | − 12·5-s + 76·9-s − 142·25-s + 632·29-s − 680·41-s − 912·45-s + 1.34e3·49-s + 328·61-s + 2.87e3·81-s − 24·89-s + 4.05e3·101-s − 8.05e3·109-s − 1.61e3·121-s + 3.63e3·125-s + 127-s + 131-s + 137-s + 139-s − 7.58e3·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + 7.86e3·169-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | − 1.07·5-s + 2.81·9-s − 1.13·25-s + 4.04·29-s − 2.59·41-s − 3.02·45-s + 3.90·49-s + 0.688·61-s + 3.94·81-s − 0.0285·89-s + 3.99·101-s − 7.07·109-s − 1.21·121-s + 2.60·125-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s − 4.34·145-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 3.57·169-s + 0.000439·173-s + 0.000417·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.298374124\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.298374124\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( ( 1 + 6 T + p^{3} T^{2} )^{2} \) |
| good | 3 | $C_2^2$ | \( ( 1 - 38 T^{2} + p^{6} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 - 670 T^{2} + p^{6} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 806 T^{2} + p^{6} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 3930 T^{2} + p^{6} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 7970 T^{2} + p^{6} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 2986 T^{2} + p^{6} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 21630 T^{2} + p^{6} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 158 T + p^{3} T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 + 29886 T^{2} + p^{6} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 22890 T^{2} + p^{6} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 170 T + p^{3} T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 59158 T^{2} + p^{6} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 148110 T^{2} + p^{6} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 52298 T^{2} + p^{6} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 6842 T^{2} + p^{6} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 82 T + p^{3} T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 122662 T^{2} + p^{6} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 182482 T^{2} + p^{6} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 592434 T^{2} + p^{6} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 + 867294 T^{2} + p^{6} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 259974 T^{2} + p^{6} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p^{3} T^{2} )^{4} \) |
| 97 | $C_2^2$ | \( ( 1 - 665346 T^{2} + p^{6} T^{4} )^{2} \) |
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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
−8.887327729557964523310200075615, −8.581015133900787638585976708754, −8.520571410223212433306103359710, −7.926000041032225876137078552307, −7.917098763018298539358160331956, −7.61275506297247953397523927158, −7.29200167871871505631574407063, −6.98260088424000844185668824631, −6.71655501786741671270772032708, −6.56257294597258001810564248767, −6.37414563291957369167499277192, −5.71813857381781443420758680905, −5.35018745590059870492832433022, −5.07792270804021463522386212568, −4.59556653044008796313367933416, −4.43679408326166539813123857555, −4.10249846907163505199746653700, −3.86938786102425746351579754493, −3.57624059174523308072536096129, −2.97719984717727987598100411914, −2.47295271190011491138840026434, −2.07465652426443788323769838947, −1.22327310670979721906030170278, −1.21550010448682081652846694696, −0.45488452933802413072555484572,
0.45488452933802413072555484572, 1.21550010448682081652846694696, 1.22327310670979721906030170278, 2.07465652426443788323769838947, 2.47295271190011491138840026434, 2.97719984717727987598100411914, 3.57624059174523308072536096129, 3.86938786102425746351579754493, 4.10249846907163505199746653700, 4.43679408326166539813123857555, 4.59556653044008796313367933416, 5.07792270804021463522386212568, 5.35018745590059870492832433022, 5.71813857381781443420758680905, 6.37414563291957369167499277192, 6.56257294597258001810564248767, 6.71655501786741671270772032708, 6.98260088424000844185668824631, 7.29200167871871505631574407063, 7.61275506297247953397523927158, 7.917098763018298539358160331956, 7.926000041032225876137078552307, 8.520571410223212433306103359710, 8.581015133900787638585976708754, 8.887327729557964523310200075615
