| L(s) = 1 | − 14·5-s − 18·9-s − 2·13-s − 96·17-s + 99·25-s + 110·29-s − 192·37-s + 14·41-s + 252·45-s − 73·49-s − 192·53-s − 50·61-s + 28·65-s − 480·73-s + 243·81-s + 1.34e3·85-s + 480·89-s − 50·97-s + 226·101-s + 768·109-s − 62·113-s + 36·117-s − 73·121-s − 714·125-s + ⋯ |
| L(s) = 1 | − 2.79·5-s − 2·9-s − 0.153·13-s − 5.64·17-s + 3.95·25-s + 3.79·29-s − 5.18·37-s + 0.341·41-s + 28/5·45-s − 1.48·49-s − 3.62·53-s − 0.819·61-s + 0.430·65-s − 6.57·73-s + 3·81-s + 15.8·85-s + 5.39·89-s − 0.515·97-s + 2.23·101-s + 7.04·109-s − 0.548·113-s + 4/13·117-s − 0.603·121-s − 5.71·125-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.1223696230\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1223696230\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| good | 5 | $C_2^2$ | \( ( 1 + 7 T + 24 T^{2} + 7 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 7 | $C_2^3$ | \( 1 + 73 T^{2} + 2928 T^{4} + 73 p^{4} T^{6} + p^{8} T^{8} \) |
| 11 | $C_2^3$ | \( 1 + 73 T^{2} - 9312 T^{4} + 73 p^{4} T^{6} + p^{8} T^{8} \) |
| 13 | $C_2$ | \( ( 1 - 22 T + p^{2} T^{2} )^{2}( 1 + 23 T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 24 T + p^{2} T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 - 146 T^{2} + p^{4} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 + 889 T^{2} + 510480 T^{4} + 889 p^{4} T^{6} + p^{8} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 - 55 T + 2184 T^{2} - 55 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2^3$ | \( 1 + 1561 T^{2} + 1513200 T^{4} + 1561 p^{4} T^{6} + p^{8} T^{8} \) |
| 37 | $C_2$ | \( ( 1 + 48 T + p^{2} T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 - 7 T - 1632 T^{2} - 7 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 43 | $C_2^3$ | \( 1 + 889 T^{2} - 2628480 T^{4} + 889 p^{4} T^{6} + p^{8} T^{8} \) |
| 47 | $C_2^3$ | \( 1 + 3049 T^{2} + 4416720 T^{4} + 3049 p^{4} T^{6} + p^{8} T^{8} \) |
| 53 | $C_2$ | \( ( 1 + 48 T + p^{2} T^{2} )^{4} \) |
| 59 | $C_2^3$ | \( 1 + 73 T^{2} - 12112032 T^{4} + 73 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 + 25 T - 3096 T^{2} + 25 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $C_2^3$ | \( 1 - 1223 T^{2} - 18655392 T^{4} - 1223 p^{4} T^{6} + p^{8} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 - 10078 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 120 T + p^{2} T^{2} )^{4} \) |
| 79 | $C_2^3$ | \( 1 - 743 T^{2} - 38398032 T^{4} - 743 p^{4} T^{6} + p^{8} T^{8} \) |
| 83 | $C_2^3$ | \( 1 + 10297 T^{2} + 58569888 T^{4} + 10297 p^{4} T^{6} + p^{8} T^{8} \) |
| 89 | $C_2$ | \( ( 1 - 120 T + p^{2} T^{2} )^{4} \) |
| 97 | $C_2^2$ | \( ( 1 + 25 T - 8784 T^{2} + 25 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
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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
−8.572074029711444926838466863491, −8.250683263237025394531218872997, −7.81445949533297943830460641278, −7.74970212450779332292464902869, −7.18378397775153678100805783971, −7.07656474968035428331201372435, −6.98684994734887161720076090581, −6.39359138833270893162149784532, −6.30346014064565940196220175001, −6.25571418057463130966171271085, −5.96009558996626181338026242445, −4.86022615659009088668919857358, −4.81702458819028046219438643134, −4.81512195059259284433252868126, −4.73988210623307925397861763784, −4.37167864626628959744513115274, −3.69161085977715354928772782006, −3.61508345525007336165582017410, −3.20709820118618004922989412400, −2.87071907659513895852811735792, −2.66563200381576189744054140919, −1.96100063647418992025950468962, −1.75777829188141285781054791348, −0.30105344860170233005618949989, −0.26372998617931427638768200583,
0.26372998617931427638768200583, 0.30105344860170233005618949989, 1.75777829188141285781054791348, 1.96100063647418992025950468962, 2.66563200381576189744054140919, 2.87071907659513895852811735792, 3.20709820118618004922989412400, 3.61508345525007336165582017410, 3.69161085977715354928772782006, 4.37167864626628959744513115274, 4.73988210623307925397861763784, 4.81512195059259284433252868126, 4.81702458819028046219438643134, 4.86022615659009088668919857358, 5.96009558996626181338026242445, 6.25571418057463130966171271085, 6.30346014064565940196220175001, 6.39359138833270893162149784532, 6.98684994734887161720076090581, 7.07656474968035428331201372435, 7.18378397775153678100805783971, 7.74970212450779332292464902869, 7.81445949533297943830460641278, 8.250683263237025394531218872997, 8.572074029711444926838466863491
