| L(s) = 1 | + 8·7-s + 16·13-s − 10·25-s − 32·37-s + 32·49-s + 128·91-s + 56·103-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 128·169-s + 173-s − 80·175-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯ |
| L(s) = 1 | + 3.02·7-s + 4.43·13-s − 2·25-s − 5.26·37-s + 32/7·49-s + 13.4·91-s + 5.51·103-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 + 9.84·169-s + 0.0760·173-s − 6.04·175-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.533630924\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.533630924\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| good | 7 | $C_2^2$ | \( ( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^3$ | \( 1 + 82 T^{4} + p^{4} T^{8} \) |
| 13 | $C_2^2$ | \( ( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 + 16 T + 128 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^3$ | \( 1 + 2722 T^{4} + p^{4} T^{8} \) |
| 43 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 47 | $C_2^2$ | \( ( 1 - 86 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 74 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^3$ | \( 1 - 878 T^{4} + p^{4} T^{8} \) |
| 61 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 67 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 73 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 89 | $C_2^3$ | \( 1 - 15518 T^{4} + p^{4} T^{8} \) |
| 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
−6.53889078164023175883575719433, −6.51778234158045449930650627730, −6.50190444980490579586768398182, −6.04643362070332428347895064793, −5.98977912848124216237595575197, −5.66002088388362004661863905068, −5.34972775812709278694710594823, −5.31355522898888937138739690189, −5.21815837402600005450359929857, −4.73527970265223480109049146475, −4.56403163020466382458687326896, −4.46601847594871413191489388810, −3.87326946974463255462457844117, −3.84605364044231014036184512600, −3.63612186421459995097202438735, −3.44756748908849115914978696017, −3.42568832698063406872758311179, −2.75283549626906864578815276818, −2.39965056854226908410061194493, −1.84044578635171724083271146122, −1.71641290487356399245716768086, −1.69120003800338443583909468310, −1.47105384809912020728087156705, −1.01811167614650883288383877307, −0.51085626317890080471003264007,
0.51085626317890080471003264007, 1.01811167614650883288383877307, 1.47105384809912020728087156705, 1.69120003800338443583909468310, 1.71641290487356399245716768086, 1.84044578635171724083271146122, 2.39965056854226908410061194493, 2.75283549626906864578815276818, 3.42568832698063406872758311179, 3.44756748908849115914978696017, 3.63612186421459995097202438735, 3.84605364044231014036184512600, 3.87326946974463255462457844117, 4.46601847594871413191489388810, 4.56403163020466382458687326896, 4.73527970265223480109049146475, 5.21815837402600005450359929857, 5.31355522898888937138739690189, 5.34972775812709278694710594823, 5.66002088388362004661863905068, 5.98977912848124216237595575197, 6.04643362070332428347895064793, 6.50190444980490579586768398182, 6.51778234158045449930650627730, 6.53889078164023175883575719433
