| L(s) = 1 | − 2·5-s − 2·13-s + 3·25-s + 2·29-s + 2·41-s − 49-s − 2·61-s + 4·65-s − 2·97-s − 2·101-s − 2·113-s − 121-s − 6·125-s + 127-s + 131-s + 137-s + 139-s − 4·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + 3·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | − 2·5-s − 2·13-s + 3·25-s + 2·29-s + 2·41-s − 49-s − 2·61-s + 4·65-s − 2·97-s − 2·101-s − 2·113-s − 121-s − 6·125-s + 127-s + 131-s + 137-s + 139-s − 4·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + 3·169-s + 173-s + 179-s + 181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3899830279\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3899830279\) |
| \(L(1)\) |
|
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_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 7 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 11 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 23 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 29 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 31 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 37 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 41 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 43 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 47 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 53 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 59 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 67 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 73 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 79 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 83 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 89 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
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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
−7.64406817390733957818753947442, −7.15978133256761823722844758022, −7.05795311548798955758465472910, −7.01676895785692401398037104609, −6.62713019005439960754788210586, −6.57402665427863424024158611834, −6.27570802743728072246292887696, −5.79422202109496432310981615385, −5.67042621352447369701036155002, −5.40865390099123074064642566586, −4.99342168953556612427332942731, −4.85322375233883259677854437046, −4.72901007213942848539095926148, −4.29677699607150867429167970752, −4.25470456806954043982600474452, −4.05484096150598663645884956589, −3.72169157809914134317815112880, −3.14463724932251819374637529893, −3.12158720315593732906587942993, −2.75698986940459864521023111760, −2.63727714629547576615179841532, −2.30605213324246940552615166060, −1.60253926696782985695749740993, −1.30070514683736634751888309100, −0.58868079306390712002911967979,
0.58868079306390712002911967979, 1.30070514683736634751888309100, 1.60253926696782985695749740993, 2.30605213324246940552615166060, 2.63727714629547576615179841532, 2.75698986940459864521023111760, 3.12158720315593732906587942993, 3.14463724932251819374637529893, 3.72169157809914134317815112880, 4.05484096150598663645884956589, 4.25470456806954043982600474452, 4.29677699607150867429167970752, 4.72901007213942848539095926148, 4.85322375233883259677854437046, 4.99342168953556612427332942731, 5.40865390099123074064642566586, 5.67042621352447369701036155002, 5.79422202109496432310981615385, 6.27570802743728072246292887696, 6.57402665427863424024158611834, 6.62713019005439960754788210586, 7.01676895785692401398037104609, 7.05795311548798955758465472910, 7.15978133256761823722844758022, 7.64406817390733957818753947442
