| L(s) = 1 | + 2·4-s − 12·9-s + 16·11-s + 16-s + 8·25-s + 32·29-s − 24·36-s + 32·44-s − 2·64-s + 64·71-s + 54·81-s − 192·99-s + 16·100-s − 48·109-s + 64·116-s + 140·121-s + 127-s + 131-s + 137-s + 139-s − 12·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 32·169-s + ⋯ |
| L(s) = 1 | + 4-s − 4·9-s + 4.82·11-s + 1/4·16-s + 8/5·25-s + 5.94·29-s − 4·36-s + 4.82·44-s − 1/4·64-s + 7.59·71-s + 6·81-s − 19.2·99-s + 8/5·100-s − 4.59·109-s + 5.94·116-s + 12.7·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 144-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(9.818981358\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.818981358\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 5 | \( 1 - 8 T^{2} + 39 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} \) |
| 7 | \( 1 \) |
| good | 3 | \( ( 1 + p T^{2} + p^{2} T^{4} )^{4} \) |
| 11 | \( ( 1 - 4 T + 5 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 13 | \( ( 1 - 8 T^{2} + p^{2} T^{4} )^{4} \) |
| 17 | \( ( 1 + 16 T^{2} - 33 T^{4} + 16 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 - 6 T^{2} - 325 T^{4} - 6 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 + p T^{2} + p^{2} T^{4} )^{4} \) |
| 29 | \( ( 1 - 4 T + p T^{2} )^{8} \) |
| 31 | \( ( 1 - 30 T^{2} - 61 T^{4} - 30 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 + 38 T^{2} + 75 T^{4} + 38 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 + 80 T^{2} + p^{2} T^{4} )^{4} \) |
| 43 | \( ( 1 + 58 T^{2} + p^{2} T^{4} )^{4} \) |
| 47 | \( ( 1 + p T^{2} + p^{2} T^{4} )^{4} \) |
| 53 | \( ( 1 - 38 T^{2} - 1365 T^{4} - 38 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 - 18 T + 167 T^{2} - 18 p T^{3} + p^{2} T^{4} )^{2}( 1 + 18 T + 167 T^{2} + 18 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | \( ( 1 - 72 T^{2} + 1463 T^{4} - 72 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 - 10 T^{2} - 4389 T^{4} - 10 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 - 8 T + p T^{2} )^{8} \) |
| 73 | \( ( 1 + 128 T^{2} + 11055 T^{4} + 128 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 - p T^{2} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 + 122 T^{2} + p^{2} T^{4} )^{4} \) |
| 89 | \( ( 1 - 160 T^{2} + 17679 T^{4} - 160 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 - 176 T^{2} + p^{2} T^{4} )^{4} \) |
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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
−4.86170995054942863367183419711, −4.71706723038461565135113715189, −4.46843895622302966768504186026, −4.33542830468016133310627071688, −4.24807527817287375369643201222, −4.08467674449296409002398588219, −3.93139444784398809007060431602, −3.79979255576366400552268071327, −3.68806037444316483404578936004, −3.52800212076404499604615505075, −3.09704937886795151950199254526, −3.09649652527565925552851786446, −3.01135547225699626004601892413, −2.82332300071326613655919052884, −2.79173113980320049660532182227, −2.78113852415835910882478118964, −2.37090541117440441318954254126, −2.06242027240983658224898531033, −2.01410231579602719698659106048, −1.84844258356382484747429837283, −1.33120615283640173607877457626, −1.32414384190304421956120807611, −0.77000627600658959443498098976, −0.76804373904933103800544173317, −0.76220086117549775617157546450,
0.76220086117549775617157546450, 0.76804373904933103800544173317, 0.77000627600658959443498098976, 1.32414384190304421956120807611, 1.33120615283640173607877457626, 1.84844258356382484747429837283, 2.01410231579602719698659106048, 2.06242027240983658224898531033, 2.37090541117440441318954254126, 2.78113852415835910882478118964, 2.79173113980320049660532182227, 2.82332300071326613655919052884, 3.01135547225699626004601892413, 3.09649652527565925552851786446, 3.09704937886795151950199254526, 3.52800212076404499604615505075, 3.68806037444316483404578936004, 3.79979255576366400552268071327, 3.93139444784398809007060431602, 4.08467674449296409002398588219, 4.24807527817287375369643201222, 4.33542830468016133310627071688, 4.46843895622302966768504186026, 4.71706723038461565135113715189, 4.86170995054942863367183419711
Plot not available for L-functions of degree greater than 10.
