| L(s) = 1 | + 4·7-s + 4·17-s − 12·23-s + 12·25-s − 8·31-s − 4·41-s + 16·47-s + 10·49-s − 20·71-s − 48·73-s + 8·79-s − 28·89-s + 32·97-s + 56·103-s + 16·113-s + 16·119-s + 36·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s − 48·161-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 1.51·7-s + 0.970·17-s − 2.50·23-s + 12/5·25-s − 1.43·31-s − 0.624·41-s + 2.33·47-s + 10/7·49-s − 2.37·71-s − 5.61·73-s + 0.900·79-s − 2.96·89-s + 3.24·97-s + 5.51·103-s + 1.50·113-s + 1.46·119-s + 3.27·121-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 − 3.78·161-s + 0.0783·163-s + 0.0773·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.347506969\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.347506969\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{4} \) |
| good | 5 | $D_4\times C_2$ | \( 1 - 12 T^{2} + 74 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 36 T^{2} + 554 T^{4} - 36 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( ( 1 - 2 T + 8 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( ( 1 + 6 T + 52 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $D_4\times C_2$ | \( 1 - 60 T^{2} + 1814 T^{4} - 60 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2}( 1 + 12 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( ( 1 + 2 T + 8 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 68 T^{2} + 4086 T^{4} - 68 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 53 | $C_2^2$ | \( ( 1 - 94 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 54 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 - 20 T^{2} + 5814 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 20 T^{2} + 2166 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 10 T + 92 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_{4}$ | \( ( 1 + 24 T + 278 T^{2} + 24 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 236 T^{2} + 25974 T^{4} - 236 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 14 T + 200 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 - 16 T + 150 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−5.93722492760101121079540103478, −5.63673186243799756555397873485, −5.58955502756901866958094168663, −5.56756806259588147343239847782, −5.18682411466633153839234183142, −4.93518315202610551654211155459, −4.69749552492762156687838958417, −4.48814935915892974003000148115, −4.40450300818983863673685508024, −4.23943905319243673550847311058, −4.18787883299432359224542559178, −3.66741921796943744819764285675, −3.48708747666927083352836818655, −3.23067072893584540880053657707, −3.07443917260044705390106669739, −2.99924864446221782492158902240, −2.57368479375650189222143078429, −2.18219372037005450272230572390, −2.07345965149007823545294692811, −1.79578379662788539949873156126, −1.72648302533497755426520917568, −1.24405212232869413154634870409, −1.06711521518020075848837160687, −0.67540059096096803958696853200, −0.29753315247689107408285330784,
0.29753315247689107408285330784, 0.67540059096096803958696853200, 1.06711521518020075848837160687, 1.24405212232869413154634870409, 1.72648302533497755426520917568, 1.79578379662788539949873156126, 2.07345965149007823545294692811, 2.18219372037005450272230572390, 2.57368479375650189222143078429, 2.99924864446221782492158902240, 3.07443917260044705390106669739, 3.23067072893584540880053657707, 3.48708747666927083352836818655, 3.66741921796943744819764285675, 4.18787883299432359224542559178, 4.23943905319243673550847311058, 4.40450300818983863673685508024, 4.48814935915892974003000148115, 4.69749552492762156687838958417, 4.93518315202610551654211155459, 5.18682411466633153839234183142, 5.56756806259588147343239847782, 5.58955502756901866958094168663, 5.63673186243799756555397873485, 5.93722492760101121079540103478
