| L(s) = 1 | + 2·4-s − 9-s − 12·11-s + 16-s + 28·19-s + 18·29-s − 20·31-s − 2·36-s + 18·41-s − 24·44-s + 52·49-s + 22·61-s − 2·64-s + 56·76-s + 14·79-s + 13·81-s + 12·99-s + 8·109-s + 36·116-s + 14·121-s − 40·124-s + 127-s + 131-s + 137-s + 139-s − 144-s + 149-s + ⋯ |
| L(s) = 1 | + 4-s − 1/3·9-s − 3.61·11-s + 1/4·16-s + 6.42·19-s + 3.34·29-s − 3.59·31-s − 1/3·36-s + 2.81·41-s − 3.61·44-s + 52/7·49-s + 2.81·61-s − 1/4·64-s + 6.42·76-s + 1.57·79-s + 13/9·81-s + 1.20·99-s + 0.766·109-s + 3.34·116-s + 1.27·121-s − 3.59·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.0833·144-s + 0.0819·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{16} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{16} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(13.30933328\) |
| \(L(\frac12)\) |
\(\approx\) |
\(13.30933328\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 5 | \( 1 \) |
| 19 | \( ( 1 - 7 T + p T^{2} )^{4} \) |
| good | 3 | \( 1 + T^{2} - 4 p T^{4} - 5 T^{6} + 79 T^{8} - 5 p^{2} T^{10} - 4 p^{5} T^{12} + p^{6} T^{14} + p^{8} T^{16} \) |
| 7 | \( ( 1 - 13 T^{2} + p^{2} T^{4} )^{4} \) |
| 11 | \( ( 1 + 3 T + 19 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{4} \) |
| 13 | \( 1 + 29 T^{2} + 424 T^{4} + 2291 T^{6} + 3199 T^{8} + 2291 p^{2} T^{10} + 424 p^{4} T^{12} + 29 p^{6} T^{14} + p^{8} T^{16} \) |
| 17 | \( 1 + 53 T^{2} + 1576 T^{4} + 34715 T^{6} + 624967 T^{8} + 34715 p^{2} T^{10} + 1576 p^{4} T^{12} + 53 p^{6} T^{14} + p^{8} T^{16} \) |
| 23 | \( ( 1 + 25 T^{2} + 96 T^{4} + 25 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 - 9 T + 8 T^{2} - 135 T^{3} + 2139 T^{4} - 135 p T^{5} + 8 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 + 5 T + 63 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{4} \) |
| 37 | \( ( 1 - 104 T^{2} + 5358 T^{4} - 104 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 - 9 T + 26 T^{2} + 243 T^{3} - 1977 T^{4} + 243 p T^{5} + 26 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 43 | \( 1 + 137 T^{2} + 10636 T^{4} + 607595 T^{6} + 28023007 T^{8} + 607595 p^{2} T^{10} + 10636 p^{4} T^{12} + 137 p^{6} T^{14} + p^{8} T^{16} \) |
| 47 | \( 1 + 2 T^{2} - 1391 T^{4} - 6046 T^{6} - 2942492 T^{8} - 6046 p^{2} T^{10} - 1391 p^{4} T^{12} + 2 p^{6} T^{14} + p^{8} T^{16} \) |
| 53 | \( ( 1 + 85 T^{2} + 4416 T^{4} + 85 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 - 97 T^{2} + 5928 T^{4} - 97 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - 11 T + 16 T^{2} + 187 T^{3} - 77 T^{4} + 187 p T^{5} + 16 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 67 | \( 1 - 16 T^{2} + 1378 T^{4} + 161600 T^{6} - 20041421 T^{8} + 161600 p^{2} T^{10} + 1378 p^{4} T^{12} - 16 p^{6} T^{14} + p^{8} T^{16} \) |
| 71 | \( ( 1 - 121 T^{2} + 9600 T^{4} - 121 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( 1 + 281 T^{2} + 48568 T^{4} + 5545535 T^{6} + 474553039 T^{8} + 5545535 p^{2} T^{10} + 48568 p^{4} T^{12} + 281 p^{6} T^{14} + p^{8} T^{16} \) |
| 79 | \( ( 1 - 7 T + 10 T^{2} + 833 T^{3} - 8591 T^{4} + 833 p T^{5} + 10 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 83 | \( ( 1 - 317 T^{2} + 38853 T^{4} - 317 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 - 157 T^{2} + 16728 T^{4} - 157 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( 1 + 233 T^{2} + 27616 T^{4} + 1830215 T^{6} + 112114447 T^{8} + 1830215 p^{2} T^{10} + 27616 p^{4} T^{12} + 233 p^{6} T^{14} + p^{8} T^{16} \) |
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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
−4.25096198358971297988592089157, −4.20459986011260021145760166174, −4.14680441371439956586378257520, −3.89707303783603207095730241442, −3.56662037305859099462501661891, −3.56374113926551913430038269705, −3.49466458255139254945401140441, −3.47342377423918864116902984329, −3.05520511643038047768294955475, −3.05217124332023068536556026473, −3.01024146138944610119644085571, −2.60134985922979979758304141841, −2.55682374188815906674356046025, −2.52908476112377962827003005296, −2.42097292790263321196050267076, −2.38957868283398017958556354883, −2.25464362890687194594138293454, −1.99394042394697031207350557019, −1.53089659550587831425513664400, −1.35487503992815534108296867836, −1.20566621167897072149547181271, −1.01161049812937356453726592593, −0.790790986160098835539034211647, −0.55359730061904770795446690490, −0.52931753622121281085655738764,
0.52931753622121281085655738764, 0.55359730061904770795446690490, 0.790790986160098835539034211647, 1.01161049812937356453726592593, 1.20566621167897072149547181271, 1.35487503992815534108296867836, 1.53089659550587831425513664400, 1.99394042394697031207350557019, 2.25464362890687194594138293454, 2.38957868283398017958556354883, 2.42097292790263321196050267076, 2.52908476112377962827003005296, 2.55682374188815906674356046025, 2.60134985922979979758304141841, 3.01024146138944610119644085571, 3.05217124332023068536556026473, 3.05520511643038047768294955475, 3.47342377423918864116902984329, 3.49466458255139254945401140441, 3.56374113926551913430038269705, 3.56662037305859099462501661891, 3.89707303783603207095730241442, 4.14680441371439956586378257520, 4.20459986011260021145760166174, 4.25096198358971297988592089157
Plot not available for L-functions of degree greater than 10.
