| L(s) = 1 | + 5·9-s + 14·11-s − 8·19-s − 5·25-s − 6·29-s + 20·31-s + 36·41-s − 3·49-s − 12·59-s + 48·61-s + 8·71-s − 34·79-s + 14·81-s + 70·99-s − 8·101-s − 10·109-s + 65·121-s − 8·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 5/3·9-s + 4.22·11-s − 1.83·19-s − 25-s − 1.11·29-s + 3.59·31-s + 5.62·41-s − 3/7·49-s − 1.56·59-s + 6.14·61-s + 0.949·71-s − 3.82·79-s + 14/9·81-s + 7.03·99-s − 0.796·101-s − 0.957·109-s + 5.90·121-s − 0.715·125-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 5^{6} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 5^{6} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.577509455\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.577509455\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + p T^{2} + 8 T^{3} + p^{2} T^{4} + p^{3} T^{6} \) |
| 7 | \( ( 1 + T^{2} )^{3} \) |
| good | 3 | \( 1 - 5 T^{2} + 11 T^{4} - 26 T^{6} + 11 p^{2} T^{8} - 5 p^{4} T^{10} + p^{6} T^{12} \) |
| 11 | \( ( 1 - 7 T + 41 T^{2} - 146 T^{3} + 41 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 13 | \( 1 - 9 T^{2} + 491 T^{4} - 2882 T^{6} + 491 p^{2} T^{8} - 9 p^{4} T^{10} + p^{6} T^{12} \) |
| 17 | \( 1 - 53 T^{2} + 1539 T^{4} - 31118 T^{6} + 1539 p^{2} T^{8} - 53 p^{4} T^{10} + p^{6} T^{12} \) |
| 19 | \( ( 1 + 4 T + 43 T^{2} + 160 T^{3} + 43 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 23 | \( 1 - 2 p T^{2} + 1887 T^{4} - 43972 T^{6} + 1887 p^{2} T^{8} - 2 p^{5} T^{10} + p^{6} T^{12} \) |
| 29 | \( ( 1 + 3 T + 15 T^{2} + 66 T^{3} + 15 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 31 | \( ( 1 - 10 T + 101 T^{2} - 540 T^{3} + 101 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 37 | \( ( 1 - 38 T^{2} + p^{2} T^{4} )^{3} \) |
| 41 | \( ( 1 - 18 T + 191 T^{2} - 1388 T^{3} + 191 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 43 | \( 1 - 178 T^{2} + 15575 T^{4} - 836124 T^{6} + 15575 p^{2} T^{8} - 178 p^{4} T^{10} + p^{6} T^{12} \) |
| 47 | \( 1 - 105 T^{2} + 6339 T^{4} - 285798 T^{6} + 6339 p^{2} T^{8} - 105 p^{4} T^{10} + p^{6} T^{12} \) |
| 53 | \( 1 - 130 T^{2} + 13655 T^{4} - 792060 T^{6} + 13655 p^{2} T^{8} - 130 p^{4} T^{10} + p^{6} T^{12} \) |
| 59 | \( ( 1 + 6 T + 99 T^{2} + 752 T^{3} + 99 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 61 | \( ( 1 - 24 T + 365 T^{2} - 3368 T^{3} + 365 p T^{4} - 24 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 67 | \( 1 - 174 T^{2} + 20567 T^{4} - 1533188 T^{6} + 20567 p^{2} T^{8} - 174 p^{4} T^{10} + p^{6} T^{12} \) |
| 71 | \( ( 1 - 4 T + 193 T^{2} - 504 T^{3} + 193 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 73 | \( 1 - 346 T^{2} + 55487 T^{4} - 5172972 T^{6} + 55487 p^{2} T^{8} - 346 p^{4} T^{10} + p^{6} T^{12} \) |
| 79 | \( ( 1 + 17 T + 205 T^{2} + 2138 T^{3} + 205 p T^{4} + 17 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( 1 - 70 T^{2} + 1179 T^{4} + 304148 T^{6} + 1179 p^{2} T^{8} - 70 p^{4} T^{10} + p^{6} T^{12} \) |
| 89 | \( ( 1 + 95 T^{2} + 464 T^{3} + 95 p T^{4} + p^{3} T^{6} )^{2} \) |
| 97 | \( 1 - 469 T^{2} + 98339 T^{4} - 12075534 T^{6} + 98339 p^{2} T^{8} - 469 p^{4} T^{10} + p^{6} T^{12} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.51388293268700745537239985374, −6.42774782643301646773797669946, −6.32782239262657766005977900302, −5.92086896655775988998790677597, −5.79582057990735480198262659015, −5.66766889283579745312532935898, −5.56880782013501661059614566022, −5.18147894339471837103817592521, −4.66450928892987930586168625508, −4.46467478359240890032201188341, −4.46192528904856440559900897225, −4.34843084016993675806262316187, −4.25260520515035561343464319951, −3.92790858080743725235610648313, −3.70514806944754872798961429427, −3.67456006060829895465512528568, −3.52768783983451687359342758566, −2.82594902690966833948318063002, −2.57075760395248597019951817067, −2.39981295589077538341173400792, −2.18492377511569047977755974177, −1.78721041338807820213409226156, −1.15659279637998945377542962898, −1.13258328751225955263532637511, −1.05356713822094753555425434045,
1.05356713822094753555425434045, 1.13258328751225955263532637511, 1.15659279637998945377542962898, 1.78721041338807820213409226156, 2.18492377511569047977755974177, 2.39981295589077538341173400792, 2.57075760395248597019951817067, 2.82594902690966833948318063002, 3.52768783983451687359342758566, 3.67456006060829895465512528568, 3.70514806944754872798961429427, 3.92790858080743725235610648313, 4.25260520515035561343464319951, 4.34843084016993675806262316187, 4.46192528904856440559900897225, 4.46467478359240890032201188341, 4.66450928892987930586168625508, 5.18147894339471837103817592521, 5.56880782013501661059614566022, 5.66766889283579745312532935898, 5.79582057990735480198262659015, 5.92086896655775988998790677597, 6.32782239262657766005977900302, 6.42774782643301646773797669946, 6.51388293268700745537239985374
Plot not available for L-functions of degree greater than 10.
