| L(s) = 1 | − 4·3-s + 2·4-s + 10·9-s − 8·12-s + 32·13-s + 9·16-s − 2·25-s − 32·27-s + 20·36-s − 128·39-s − 36·48-s + 64·52-s + 30·64-s + 32·73-s + 8·75-s − 32·79-s + 89·81-s − 64·97-s − 4·100-s − 40·103-s − 64·108-s + 40·109-s + 320·117-s − 28·121-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 2.30·3-s + 4-s + 10/3·9-s − 2.30·12-s + 8.87·13-s + 9/4·16-s − 2/5·25-s − 6.15·27-s + 10/3·36-s − 20.4·39-s − 5.19·48-s + 8.87·52-s + 15/4·64-s + 3.74·73-s + 0.923·75-s − 3.60·79-s + 89/9·81-s − 6.49·97-s − 2/5·100-s − 3.94·103-s − 6.15·108-s + 3.83·109-s + 29.5·117-s − 2.54·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.943643926\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.943643926\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( ( 1 + 2 T + T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 5 | \( 1 + 2 T^{2} - 21 T^{4} + 2 p^{2} T^{6} + p^{4} T^{8} \) |
| 7 | \( 1 \) |
| good | 2 | \( ( 1 - T^{2} - 3 T^{4} - p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 - 6 T + 25 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2}( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | \( ( 1 - 4 T + p T^{2} )^{8} \) |
| 17 | \( ( 1 + 26 T^{2} + 387 T^{4} + 26 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 + p T^{2} + p^{2} T^{4} )^{4} \) |
| 23 | \( ( 1 - 34 T^{2} + 627 T^{4} - 34 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{4} \) |
| 31 | \( ( 1 - 34 T^{2} + 195 T^{4} - 34 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 + p T^{2} + p^{2} T^{4} )^{4} \) |
| 41 | \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{4} \) |
| 43 | \( ( 1 - 62 T^{2} + p^{2} T^{4} )^{4} \) |
| 47 | \( ( 1 + 86 T^{2} + 5187 T^{4} + 86 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 - p T^{2} + p^{2} T^{4} )^{4} \) |
| 59 | \( ( 1 - 70 T^{2} + 1419 T^{4} - 70 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 + 26 T^{2} - 3045 T^{4} + 26 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 + 110 T^{2} + 7611 T^{4} + 110 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 - 134 T^{2} + p^{2} T^{4} )^{4} \) |
| 73 | \( ( 1 - 8 T - 9 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{4} \) |
| 79 | \( ( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 - 18 T + p T^{2} )^{4}( 1 + 18 T + p T^{2} )^{4} \) |
| 89 | \( ( 1 - 70 T^{2} - 3021 T^{4} - 70 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 + 8 T + p T^{2} )^{8} \) |
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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
−4.48255227919687108077629007726, −4.37757962405988752325907230917, −4.09475637191309815281492426865, −4.07180543743416624306869501127, −3.81679723670813480968217416315, −3.72913146452092540499999678467, −3.67518180464445118210757641545, −3.59499619455327732309988971279, −3.54039697137990367852771823310, −3.50046962140128672514401465151, −3.35035653529814721748392111824, −3.13166526079160935753840658232, −2.61180073527683494435193705060, −2.59361735484706029040392904479, −2.43847119916631803215484171248, −2.40945992274346411737221462628, −1.80217080287764123918181767626, −1.71761675171600041691113280698, −1.50339947203097472206504488283, −1.37530460747743157402968343138, −1.37401140284573142477393359906, −1.07237799576244200151087470923, −1.02125756682799015609119500035, −0.971451510419160897582969466562, −0.21897463438967241997858508972,
0.21897463438967241997858508972, 0.971451510419160897582969466562, 1.02125756682799015609119500035, 1.07237799576244200151087470923, 1.37401140284573142477393359906, 1.37530460747743157402968343138, 1.50339947203097472206504488283, 1.71761675171600041691113280698, 1.80217080287764123918181767626, 2.40945992274346411737221462628, 2.43847119916631803215484171248, 2.59361735484706029040392904479, 2.61180073527683494435193705060, 3.13166526079160935753840658232, 3.35035653529814721748392111824, 3.50046962140128672514401465151, 3.54039697137990367852771823310, 3.59499619455327732309988971279, 3.67518180464445118210757641545, 3.72913146452092540499999678467, 3.81679723670813480968217416315, 4.07180543743416624306869501127, 4.09475637191309815281492426865, 4.37757962405988752325907230917, 4.48255227919687108077629007726
Plot not available for L-functions of degree greater than 10.
