| L(s) = 1 | + 8·41-s + 8·53-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-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 + 241-s + 251-s − 256-s + ⋯ |
| L(s) = 1 | + 8·41-s + 8·53-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-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 + 241-s + 251-s − 256-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 5^{8} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 5^{8} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(3.339359679\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.339359679\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T^{8} \) |
| 3 | \( 1 \) |
| 5 | \( ( 1 + T^{4} )^{2} \) |
| 17 | \( ( 1 + T^{4} )^{2} \) |
| good | 7 | \( 1 + T^{16} \) |
| 11 | \( 1 + T^{16} \) |
| 13 | \( ( 1 + T^{8} )^{2} \) |
| 19 | \( ( 1 + T^{8} )^{2} \) |
| 23 | \( 1 + T^{16} \) |
| 29 | \( ( 1 + T^{2} )^{4}( 1 + T^{8} ) \) |
| 31 | \( 1 + T^{16} \) |
| 37 | \( ( 1 + T^{4} )^{2}( 1 + T^{8} ) \) |
| 41 | \( ( 1 - T )^{8}( 1 + T^{8} ) \) |
| 43 | \( ( 1 + T^{8} )^{2} \) |
| 47 | \( ( 1 - T )^{8}( 1 + T )^{8} \) |
| 53 | \( ( 1 - T )^{8}( 1 + T^{4} )^{2} \) |
| 59 | \( ( 1 + T^{8} )^{2} \) |
| 61 | \( ( 1 + T^{4} )^{2}( 1 + T^{8} ) \) |
| 67 | \( ( 1 + T^{4} )^{4} \) |
| 71 | \( 1 + T^{16} \) |
| 73 | \( ( 1 + T^{2} )^{4}( 1 + T^{8} ) \) |
| 79 | \( 1 + T^{16} \) |
| 83 | \( ( 1 + T^{8} )^{2} \) |
| 89 | \( ( 1 + T^{8} )^{2} \) |
| 97 | \( ( 1 + T^{4} )^{2}( 1 + T^{8} ) \) |
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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
−3.80843342978064287638362218088, −3.79733659762267152469941980654, −3.74636269602205829617048382713, −3.54831263977768224826002662283, −3.36712826645639734729542876990, −3.34211087911783598898137738079, −3.04445502221665069650022295393, −2.84512240431930874843005376852, −2.71974855998999049626091991739, −2.71594834573755052567038832889, −2.69237166317470861959956196399, −2.51188199420575308301051513277, −2.39930222484867921711165941502, −2.33254146379107986195832380352, −2.27126912485487564034838970679, −2.05056583119650083073397230960, −1.86533020243355741922817722702, −1.76136148501983299786157563426, −1.57349288976865048174089024217, −1.23710129093163576262541804817, −1.08181661553831789882355618683, −1.05510324335886166047219386318, −0.866295009597593236509781723983, −0.67564980425254987166418547586, −0.66157347538974979184424999251,
0.66157347538974979184424999251, 0.67564980425254987166418547586, 0.866295009597593236509781723983, 1.05510324335886166047219386318, 1.08181661553831789882355618683, 1.23710129093163576262541804817, 1.57349288976865048174089024217, 1.76136148501983299786157563426, 1.86533020243355741922817722702, 2.05056583119650083073397230960, 2.27126912485487564034838970679, 2.33254146379107986195832380352, 2.39930222484867921711165941502, 2.51188199420575308301051513277, 2.69237166317470861959956196399, 2.71594834573755052567038832889, 2.71974855998999049626091991739, 2.84512240431930874843005376852, 3.04445502221665069650022295393, 3.34211087911783598898137738079, 3.36712826645639734729542876990, 3.54831263977768224826002662283, 3.74636269602205829617048382713, 3.79733659762267152469941980654, 3.80843342978064287638362218088
Plot not available for L-functions of degree greater than 10.
