| L(s) = 1 | + 2·4-s + 8·7-s − 16·13-s − 64·19-s − 32·25-s + 16·28-s − 88·31-s − 136·37-s + 80·43-s + 114·49-s − 32·52-s − 100·61-s − 8·64-s − 16·67-s − 64·73-s − 128·76-s + 152·79-s − 128·91-s − 352·97-s − 64·100-s + 56·103-s + 224·109-s + 46·121-s − 176·124-s + 127-s + 131-s − 512·133-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 8/7·7-s − 1.23·13-s − 3.36·19-s − 1.27·25-s + 4/7·28-s − 2.83·31-s − 3.67·37-s + 1.86·43-s + 2.32·49-s − 0.615·52-s − 1.63·61-s − 1/8·64-s − 0.238·67-s − 0.876·73-s − 1.68·76-s + 1.92·79-s − 1.40·91-s − 3.62·97-s − 0.639·100-s + 0.543·103-s + 2.05·109-s + 0.380·121-s − 1.41·124-s + 0.00787·127-s + 0.00763·131-s − 3.84·133-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.6251787185\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6251787185\) |
| \(L(2)\) |
|
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 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^3$ | \( 1 + 32 T^{2} + 399 T^{4} + 32 p^{4} T^{6} + p^{8} T^{8} \) |
| 7 | $C_2^2$ | \( ( 1 - 4 T - 33 T^{2} - 4 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 11 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 14 T + 75 T^{2} - 14 p^{2} T^{3} + p^{4} T^{4} )( 1 + 14 T + 75 T^{2} + 14 p^{2} T^{3} + p^{4} T^{4} ) \) |
| 13 | $C_2^2$ | \( ( 1 + 8 T - 105 T^{2} + 8 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 416 T^{2} + p^{4} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 16 T + p^{2} T^{2} )^{4} \) |
| 23 | $C_2^3$ | \( 1 + 770 T^{2} + 313059 T^{4} + 770 p^{4} T^{6} + p^{8} T^{8} \) |
| 29 | $C_2^3$ | \( 1 + 1664 T^{2} + 2061615 T^{4} + 1664 p^{4} T^{6} + p^{8} T^{8} \) |
| 31 | $C_2^2$ | \( ( 1 + 44 T + 975 T^{2} + 44 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 34 T + p^{2} T^{2} )^{4} \) |
| 41 | $C_2^3$ | \( 1 + 1184 T^{2} - 1423905 T^{4} + 1184 p^{4} T^{6} + p^{8} T^{8} \) |
| 43 | $C_2^2$ | \( ( 1 - 40 T - 249 T^{2} - 40 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 - 2782 T^{2} + 2859843 T^{4} - 2782 p^{4} T^{6} + p^{8} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 - 4160 T^{2} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^3$ | \( 1 + 5810 T^{2} + 21638739 T^{4} + 5810 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 + 50 T - 1221 T^{2} + 50 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + 8 T - 4425 T^{2} + 8 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 7490 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 16 T + p^{2} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - 76 T - 465 T^{2} - 76 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 - 334 T^{2} - 47346765 T^{4} - 334 p^{4} T^{6} + p^{8} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 - 15680 T^{2} + p^{4} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 + 176 T + 21567 T^{2} + 176 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
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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
−9.110434110478828292079559110806, −8.727287624815488222493534994112, −8.717374288428360639008742618293, −8.526418836284378476621410743676, −8.056280778261669263405040850989, −7.81796861014634595473043464083, −7.44075688452594573551872043750, −7.20640416515618152299617012085, −7.00257818445030685737589840380, −6.90486468172323226368728828864, −6.20889367143516985852749912517, −6.07535911693213322830452563462, −5.66453604210382631405591824341, −5.52392317562612939667571792801, −4.97318677650644879623271920543, −4.82377070063963437675771575990, −4.33724813874447706699116597336, −3.98989440783965587342884810830, −3.85086579071850667810935227774, −3.28770246934692631876822675452, −2.62272613116582676311546725858, −2.16674850045852790685693800677, −1.79964609406910535143879502568, −1.79877935892192256795035608831, −0.24015488036411555722703447758,
0.24015488036411555722703447758, 1.79877935892192256795035608831, 1.79964609406910535143879502568, 2.16674850045852790685693800677, 2.62272613116582676311546725858, 3.28770246934692631876822675452, 3.85086579071850667810935227774, 3.98989440783965587342884810830, 4.33724813874447706699116597336, 4.82377070063963437675771575990, 4.97318677650644879623271920543, 5.52392317562612939667571792801, 5.66453604210382631405591824341, 6.07535911693213322830452563462, 6.20889367143516985852749912517, 6.90486468172323226368728828864, 7.00257818445030685737589840380, 7.20640416515618152299617012085, 7.44075688452594573551872043750, 7.81796861014634595473043464083, 8.056280778261669263405040850989, 8.526418836284378476621410743676, 8.717374288428360639008742618293, 8.727287624815488222493534994112, 9.110434110478828292079559110806
