| L(s) = 1 | − 44·5-s + 92·13-s + 384·17-s + 968·25-s − 412·29-s + 300·37-s − 68·49-s − 1.30e3·53-s + 156·61-s − 4.04e3·65-s + 1.26e3·81-s − 1.68e4·85-s + 1.85e3·97-s − 3.79e3·101-s − 2.33e3·109-s − 4.47e3·113-s − 1.61e4·125-s + 127-s + 131-s + 137-s + 139-s + 1.81e4·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | − 3.93·5-s + 1.96·13-s + 5.47·17-s + 7.74·25-s − 2.63·29-s + 1.33·37-s − 0.198·49-s − 3.36·53-s + 0.327·61-s − 7.72·65-s + 1.73·81-s − 21.5·85-s + 1.94·97-s − 3.73·101-s − 2.04·109-s − 3.72·113-s − 11.5·125-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 10.3·145-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.709168565\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.709168565\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| good | 3 | $C_2^3$ | \( 1 - 1262 T^{4} + p^{12} T^{8} \) |
| 5 | $C_2^2$ | \( ( 1 + 22 T + 242 T^{2} + 22 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 + 34 T^{2} + p^{6} T^{4} )^{2} \) |
| 11 | $C_2^3$ | \( 1 - 3050318 T^{4} + p^{12} T^{8} \) |
| 13 | $C_2^2$ | \( ( 1 - 46 T + 1058 T^{2} - 46 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 96 T + p^{3} T^{2} )^{4} \) |
| 19 | $C_2^3$ | \( 1 - 89434798 T^{4} + p^{12} T^{8} \) |
| 23 | $C_2^2$ | \( ( 1 - 10814 T^{2} + p^{6} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 206 T + 21218 T^{2} + 206 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 20862 T^{2} + p^{6} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 150 T + 11250 T^{2} - 150 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 40498 T^{2} + p^{6} T^{4} )^{2} \) |
| 43 | $C_2^3$ | \( 1 - 11798591182 T^{4} + p^{12} T^{8} \) |
| 47 | $C_2^2$ | \( ( 1 + 199646 T^{2} + p^{6} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 650 T + 211250 T^{2} + 650 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 59 | $C_2^3$ | \( 1 - 878 p^{4} T^{4} + p^{12} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 - 78 T + 3042 T^{2} - 78 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 67 | $C_2^3$ | \( 1 - 123064007662 T^{4} + p^{12} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 - 581342 T^{2} + p^{6} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + 517010 T^{2} + p^{6} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 83 | $C_2^3$ | \( 1 - 298648058542 T^{4} + p^{12} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 + 578162 T^{2} + p^{6} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 464 T + p^{3} T^{2} )^{4} \) |
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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
−7.64535564168918625356929155930, −7.45457091214881293531960383679, −7.36262623896528836110991739681, −6.53432318621095343605342533876, −6.51496059447431769405906220425, −6.45739200047669362457487467224, −5.77764011744220118923295571334, −5.76107694988434670238034038822, −5.27319098637975968427015801394, −5.21041150704074062722617259788, −5.11415009297821671833670464159, −4.39753156751527706738109482508, −4.05459464157130533102563451540, −3.95575408382737790645054972011, −3.94130864180529585467066875575, −3.49460106834891121801781020897, −3.27947815127603608493137909478, −3.12403143537774558805358677609, −3.10274139568286316139319854907, −2.33653512108804285997846456205, −1.32443342311444208329498893850, −1.29588081089866431592655486336, −1.28900103497461162238080410108, −0.45318681212090093734765084899, −0.36034207165489900106262885767,
0.36034207165489900106262885767, 0.45318681212090093734765084899, 1.28900103497461162238080410108, 1.29588081089866431592655486336, 1.32443342311444208329498893850, 2.33653512108804285997846456205, 3.10274139568286316139319854907, 3.12403143537774558805358677609, 3.27947815127603608493137909478, 3.49460106834891121801781020897, 3.94130864180529585467066875575, 3.95575408382737790645054972011, 4.05459464157130533102563451540, 4.39753156751527706738109482508, 5.11415009297821671833670464159, 5.21041150704074062722617259788, 5.27319098637975968427015801394, 5.76107694988434670238034038822, 5.77764011744220118923295571334, 6.45739200047669362457487467224, 6.51496059447431769405906220425, 6.53432318621095343605342533876, 7.36262623896528836110991739681, 7.45457091214881293531960383679, 7.64535564168918625356929155930
