| L(s) = 1 | − 12·5-s + 12·13-s + 12·17-s + 74·25-s − 24·29-s + 6·41-s + 14·49-s + 24·61-s − 144·65-s − 28·73-s − 144·85-s + 24·89-s − 2·97-s − 12·101-s + 12·113-s − 13·121-s − 312·125-s + 127-s + 131-s + 137-s + 139-s + 288·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | − 5.36·5-s + 3.32·13-s + 2.91·17-s + 74/5·25-s − 4.45·29-s + 0.937·41-s + 2·49-s + 3.07·61-s − 17.8·65-s − 3.27·73-s − 15.6·85-s + 2.54·89-s − 0.203·97-s − 1.19·101-s + 1.12·113-s − 1.18·121-s − 27.9·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 23.9·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.193450728\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.193450728\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( ( 1 + 6 T + 17 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^3$ | \( 1 + 13 T^{2} + 48 T^{4} + 13 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2^2$ | \( ( 1 - 6 T + 25 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 + 11 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 - 34 T^{2} + 627 T^{4} - 34 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 + 12 T + 77 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^3$ | \( 1 - 14 T^{2} - 765 T^{4} - 14 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $C_2^2$ | \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 3 T - 32 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )( 1 + 83 T^{2} + p^{2} T^{4} ) \) |
| 47 | $C_2^3$ | \( 1 - 82 T^{2} + 4515 T^{4} - 82 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 + 86 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^3$ | \( 1 + 37 T^{2} - 2112 T^{4} + 37 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2}( 1 + T + p T^{2} )^{2} \) |
| 67 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 13 T^{2} + p^{2} T^{4} )( 1 + 122 T^{2} + p^{2} T^{4} ) \) |
| 71 | $C_2^2$ | \( ( 1 + 130 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{4} \) |
| 79 | $C_2^2$$\times$$C_2^2$ | \( ( 1 + 11 T^{2} + p^{2} T^{4} )( 1 + 131 T^{2} + p^{2} T^{4} ) \) |
| 83 | $C_2^3$ | \( 1 + 22 T^{2} - 6405 T^{4} + 22 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 97 | $C_2^2$ | \( ( 1 + T - 96 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−6.88486525923135144294000271077, −6.37508568172537327865282184951, −6.27382922009887181472077184050, −5.85691140729726789168465394611, −5.75801459266505386292307701985, −5.67865060118510864468198070801, −5.30105525502554537794950382137, −5.13609737631337688428909016714, −4.86541013581821821110254887207, −4.31248681626065609538555054311, −4.24968357373400134267176830141, −4.00686884428208395189625779842, −3.83618896922361921164306969380, −3.82124569629653266151060229259, −3.67672287093841869546960577420, −3.43642642159275953340810940207, −3.29458937088407859514302507468, −2.92463521199666841935403540814, −2.81469080456095380321609080984, −2.07831428033174111454524240100, −1.62450947776392018348328545002, −1.37533795947181610441442893151, −0.843622059593794011865724141001, −0.65728010947734261886232813411, −0.38829608182196922927670084808,
0.38829608182196922927670084808, 0.65728010947734261886232813411, 0.843622059593794011865724141001, 1.37533795947181610441442893151, 1.62450947776392018348328545002, 2.07831428033174111454524240100, 2.81469080456095380321609080984, 2.92463521199666841935403540814, 3.29458937088407859514302507468, 3.43642642159275953340810940207, 3.67672287093841869546960577420, 3.82124569629653266151060229259, 3.83618896922361921164306969380, 4.00686884428208395189625779842, 4.24968357373400134267176830141, 4.31248681626065609538555054311, 4.86541013581821821110254887207, 5.13609737631337688428909016714, 5.30105525502554537794950382137, 5.67865060118510864468198070801, 5.75801459266505386292307701985, 5.85691140729726789168465394611, 6.27382922009887181472077184050, 6.37508568172537327865282184951, 6.88486525923135144294000271077
