| L(s) = 1 | + 6·4-s + 17·16-s + 36·19-s − 32·31-s − 4·49-s + 20·61-s + 42·64-s + 216·76-s + 28·79-s + 20·109-s − 76·121-s − 192·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 58·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | + 3·4-s + 17/4·16-s + 8.25·19-s − 5.74·31-s − 4/7·49-s + 2.56·61-s + 21/4·64-s + 24.7·76-s + 3.15·79-s + 1.91·109-s − 6.90·121-s − 17.2·124-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 + 4.46·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(28.20376032\) |
| \(L(\frac12)\) |
\(\approx\) |
\(28.20376032\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - 34 T^{4} + p^{4} T^{8} \) |
| 7 | \( ( 1 + T^{2} )^{4} \) |
| good | 2 | \( ( 1 - T - T^{2} - p T^{3} + p^{2} T^{4} )^{2}( 1 + T - T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | \( ( 1 + 19 T^{2} + p^{2} T^{4} )^{4} \) |
| 13 | \( ( 1 - 29 T^{2} + 417 T^{4} - 29 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 - 27 T^{2} + 37 p T^{4} - 27 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 - 9 T + 53 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{4} \) |
| 23 | \( ( 1 - 59 T^{2} + 1881 T^{4} - 59 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 + 75 T^{2} + 2957 T^{4} + 75 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 + 8 T + 57 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{4} \) |
| 37 | \( ( 1 + 22 T^{2} + 2523 T^{4} + 22 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 + 39 T^{2} + 3317 T^{4} + 39 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 - 85 T^{2} + p^{2} T^{4} )^{4} \) |
| 47 | \( ( 1 - 91 T^{2} + p^{2} T^{4} )^{4} \) |
| 53 | \( ( 1 - 167 T^{2} + 12165 T^{4} - 167 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 + 6 T^{2} + 3947 T^{4} + 6 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - 5 T + 123 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{4} \) |
| 67 | \( ( 1 - 113 T^{2} + 6453 T^{4} - 113 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 + 75 T^{2} + 857 T^{4} + 75 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 - 125 T^{2} + p^{2} T^{4} )^{4} \) |
| 79 | \( ( 1 - 7 T + 123 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 + 18 T^{2} - 2605 T^{4} + 18 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 + 103 T^{2} + p^{2} T^{4} )^{4} \) |
| 97 | \( ( 1 - 233 T^{2} + 26673 T^{4} - 233 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
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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
−4.26669587787356413789982273579, −3.98997775338137963929094197101, −3.90485548420766991901229049357, −3.81625389619872700209845439160, −3.72503868177621910729527527071, −3.61145116036443944254329650659, −3.37335262069966511475326813110, −3.26175817970916382456223331011, −3.24131273684592580492800073099, −3.06679691119750474647143765210, −2.94853352910896821568089382637, −2.92243670996024171884095627553, −2.77740693530013879583669126319, −2.50707273831073543360774532949, −2.21497719604253992350322779417, −2.13676905240918307572456650368, −1.90916921526486606349985009881, −1.89713942362452134568231293296, −1.74186733076079139804656394510, −1.60883210237054922634982649309, −1.30559380936349052883061588368, −0.972420488616498831440526230080, −0.902267585546710843733223853518, −0.878488367702562355798360889511, −0.41028041964600514500615788575,
0.41028041964600514500615788575, 0.878488367702562355798360889511, 0.902267585546710843733223853518, 0.972420488616498831440526230080, 1.30559380936349052883061588368, 1.60883210237054922634982649309, 1.74186733076079139804656394510, 1.89713942362452134568231293296, 1.90916921526486606349985009881, 2.13676905240918307572456650368, 2.21497719604253992350322779417, 2.50707273831073543360774532949, 2.77740693530013879583669126319, 2.92243670996024171884095627553, 2.94853352910896821568089382637, 3.06679691119750474647143765210, 3.24131273684592580492800073099, 3.26175817970916382456223331011, 3.37335262069966511475326813110, 3.61145116036443944254329650659, 3.72503868177621910729527527071, 3.81625389619872700209845439160, 3.90485548420766991901229049357, 3.98997775338137963929094197101, 4.26669587787356413789982273579
Plot not available for L-functions of degree greater than 10.
