L(s) = 1 | − 4·9-s + 8·25-s − 64·41-s − 32·49-s − 32·73-s + 10·81-s − 16·89-s + 64·97-s + 144·113-s + 24·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 32·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
L(s) = 1 | − 4/3·9-s + 8/5·25-s − 9.99·41-s − 4.57·49-s − 3.74·73-s + 10/9·81-s − 1.69·89-s + 6.49·97-s + 13.5·113-s + 2.18·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2.46·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{80} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{80} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.08295286642\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.08295286642\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( ( 1 + T^{2} )^{4} \) |
good | 5 | \( ( 1 - 4 T^{2} + 6 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 7 | \( ( 1 + 8 T^{2} + p^{2} T^{4} )^{4} \) |
| 11 | \( ( 1 - 12 T^{2} + 86 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 - 8 T^{2} + p^{2} T^{4} )^{4} \) |
| 17 | \( ( 1 + 22 T^{2} + p^{2} T^{4} )^{4} \) |
| 19 | \( ( 1 - 20 T^{2} + 54 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{4} \) |
| 29 | \( ( 1 - 4 T^{2} + 1254 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 + 8 T^{2} + p^{2} T^{4} )^{4} \) |
| 37 | \( ( 1 - 64 T^{2} + 2034 T^{4} - 64 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 + 16 T + 134 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{4} \) |
| 43 | \( ( 1 - 74 T^{2} + p^{2} T^{4} )^{4} \) |
| 47 | \( ( 1 + 86 T^{2} + p^{2} T^{4} )^{4} \) |
| 53 | \( ( 1 - 100 T^{2} + 6918 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 + 74 T^{2} + p^{2} T^{4} )^{4} \) |
| 61 | \( ( 1 - 160 T^{2} + 12114 T^{4} - 160 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 - 44 T^{2} - 2826 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 - 20 T^{2} - 2106 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 + 4 T + p T^{2} )^{8} \) |
| 79 | \( ( 1 + 152 T^{2} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 - 300 T^{2} + 36086 T^{4} - 300 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 + 4 T + 134 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 97 | \( ( 1 - 16 T + 210 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.57050508329634882885912643450, −3.49493296933399153732717122050, −3.28606272789826592172688168510, −3.23565513698055294549827711682, −3.20393517923070790941502697632, −3.15238176684994368769276575086, −3.00525924559756471847679565562, −2.85184368470712548972350310692, −2.73914317915586154752965924193, −2.72733274163521646050720287110, −2.15104830739903089818967318878, −2.14604808605590729457305085554, −2.04575131314617586948965209839, −2.02396068775398714000275574176, −1.94488714456275599405769581445, −1.71374744279873632203365342915, −1.66014394842254437953384780513, −1.54093108087126727697522467868, −1.38122507721193970393332904551, −1.09096439818660904477654821370, −1.00880910509107658070971662147, −0.54300852288144398598177176643, −0.49078491704823333932801931419, −0.42095362951038571144898499439, −0.03022247252965957023753620645,
0.03022247252965957023753620645, 0.42095362951038571144898499439, 0.49078491704823333932801931419, 0.54300852288144398598177176643, 1.00880910509107658070971662147, 1.09096439818660904477654821370, 1.38122507721193970393332904551, 1.54093108087126727697522467868, 1.66014394842254437953384780513, 1.71374744279873632203365342915, 1.94488714456275599405769581445, 2.02396068775398714000275574176, 2.04575131314617586948965209839, 2.14604808605590729457305085554, 2.15104830739903089818967318878, 2.72733274163521646050720287110, 2.73914317915586154752965924193, 2.85184368470712548972350310692, 3.00525924559756471847679565562, 3.15238176684994368769276575086, 3.20393517923070790941502697632, 3.23565513698055294549827711682, 3.28606272789826592172688168510, 3.49493296933399153732717122050, 3.57050508329634882885912643450
Plot not available for L-functions of degree greater than 10.