| L(s) = 1 | + 24·25-s − 64·47-s + 12·49-s − 80·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 104·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + ⋯ |
| L(s) = 1 | + 24/5·25-s − 9.33·47-s + 12/7·49-s − 7.27·121-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 − 8·169-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 + 0.0688·211-s + 0.0669·223-s + 0.0663·227-s + 0.0660·229-s + 0.0655·233-s + 0.0646·239-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.08035783631\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.08035783631\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{2} \) |
| good | 5 | \( ( 1 - 4 T + p T^{2} )^{4}( 1 + 4 T + p T^{2} )^{4} \) |
| 11 | \( ( 1 + 20 T^{2} + p^{2} T^{4} )^{4} \) |
| 13 | \( ( 1 + p T^{2} )^{8} \) |
| 17 | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{4} \) |
| 19 | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{4} \) |
| 23 | \( ( 1 - 36 T^{2} + p^{2} T^{4} )^{4} \) |
| 29 | \( ( 1 + 48 T^{2} + p^{2} T^{4} )^{4} \) |
| 31 | \( ( 1 - 30 T^{2} + p^{2} T^{4} )^{4} \) |
| 37 | \( ( 1 - 54 T^{2} + p^{2} T^{4} )^{4} \) |
| 41 | \( ( 1 + 62 T^{2} + p^{2} T^{4} )^{4} \) |
| 43 | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{4} \) |
| 47 | \( ( 1 + 8 T + p T^{2} )^{8} \) |
| 53 | \( ( 1 + 16 T^{2} + p^{2} T^{4} )^{4} \) |
| 59 | \( ( 1 - 38 T^{2} + p^{2} T^{4} )^{4} \) |
| 61 | \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{4} \) |
| 67 | \( ( 1 - 130 T^{2} + p^{2} T^{4} )^{4} \) |
| 71 | \( ( 1 - 52 T^{2} + p^{2} T^{4} )^{4} \) |
| 73 | \( ( 1 - 106 T^{2} + p^{2} T^{4} )^{4} \) |
| 79 | \( ( 1 + 78 T^{2} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 - p T^{2} )^{8} \) |
| 89 | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{4} \) |
| 97 | \( ( 1 - 154 T^{2} + p^{2} T^{4} )^{4} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.45554915536301805792665951306, −3.38413847533821488868943838182, −3.30904915505275517370560747893, −3.13335864231904509362101276878, −3.02133000744937370686836278481, −2.85836646408021918306616744900, −2.72200236696222066444856604833, −2.67664573973429509054950427549, −2.62883721522547116458857448311, −2.50650010967701048603588579740, −2.41663990797842577958045699903, −2.11129268161478995381942462962, −2.08056496632941336419480641476, −1.92235177518961962412563699462, −1.76358509724065347547878651284, −1.45720189999462311747470159930, −1.38293303929492346929253564356, −1.33470196896476974794399842610, −1.31456706267932608709816729092, −1.15775240339099621591368804494, −1.03579387247350341215705336896, −0.77627007633338196211317028281, −0.49388014411569350700394108157, −0.17012476403075167705947469838, −0.03971361901561717906303997567,
0.03971361901561717906303997567, 0.17012476403075167705947469838, 0.49388014411569350700394108157, 0.77627007633338196211317028281, 1.03579387247350341215705336896, 1.15775240339099621591368804494, 1.31456706267932608709816729092, 1.33470196896476974794399842610, 1.38293303929492346929253564356, 1.45720189999462311747470159930, 1.76358509724065347547878651284, 1.92235177518961962412563699462, 2.08056496632941336419480641476, 2.11129268161478995381942462962, 2.41663990797842577958045699903, 2.50650010967701048603588579740, 2.62883721522547116458857448311, 2.67664573973429509054950427549, 2.72200236696222066444856604833, 2.85836646408021918306616744900, 3.02133000744937370686836278481, 3.13335864231904509362101276878, 3.30904915505275517370560747893, 3.38413847533821488868943838182, 3.45554915536301805792665951306
Plot not available for L-functions of degree greater than 10.
