| L(s) = 1 | − 3·9-s + 4·19-s + 25-s + 18·29-s − 26·31-s − 10·41-s + 39·49-s − 10·59-s − 40·61-s − 14·71-s + 8·79-s + 6·81-s − 4·89-s − 38·101-s + 4·109-s − 58·121-s + 16·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 46·169-s + ⋯ |
| L(s) = 1 | − 9-s + 0.917·19-s + 1/5·25-s + 3.34·29-s − 4.66·31-s − 1.56·41-s + 39/7·49-s − 1.30·59-s − 5.12·61-s − 1.66·71-s + 0.900·79-s + 2/3·81-s − 0.423·89-s − 3.78·101-s + 0.383·109-s − 5.27·121-s + 1.43·125-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 + 3.53·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{6} \cdot 5^{6} \cdot 23^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{6} \cdot 5^{6} \cdot 23^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3215195155\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3215195155\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 + T^{2} )^{3} \) |
| 5 | \( 1 - T^{2} - 16 T^{3} - p T^{4} + p^{3} T^{6} \) |
| 23 | \( ( 1 + T^{2} )^{3} \) |
| good | 7 | \( ( 1 - 13 T^{2} + p^{2} T^{4} )^{3} \) |
| 11 | \( ( 1 + 29 T^{2} - 2 T^{3} + 29 p T^{4} + p^{3} T^{6} )^{2} \) |
| 13 | \( 1 - 46 T^{2} + 79 p T^{4} - 15448 T^{6} + 79 p^{3} T^{8} - 46 p^{4} T^{10} + p^{6} T^{12} \) |
| 17 | \( 1 - 71 T^{2} + 2502 T^{4} - 53231 T^{6} + 2502 p^{2} T^{8} - 71 p^{4} T^{10} + p^{6} T^{12} \) |
| 19 | \( ( 1 - 2 T + 43 T^{2} - 86 T^{3} + 43 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 29 | \( ( 1 - 9 T + 84 T^{2} - 461 T^{3} + 84 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 31 | \( ( 1 + 13 T + 144 T^{2} + 865 T^{3} + 144 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 37 | \( 1 - 163 T^{2} + 12278 T^{4} - 562959 T^{6} + 12278 p^{2} T^{8} - 163 p^{4} T^{10} + p^{6} T^{12} \) |
| 41 | \( ( 1 + 5 T + 66 T^{2} + 435 T^{3} + 66 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 43 | \( 1 - 178 T^{2} + 15575 T^{4} - 834444 T^{6} + 15575 p^{2} T^{8} - 178 p^{4} T^{10} + p^{6} T^{12} \) |
| 47 | \( 1 - 190 T^{2} + 16919 T^{4} - 955248 T^{6} + 16919 p^{2} T^{8} - 190 p^{4} T^{10} + p^{6} T^{12} \) |
| 53 | \( 1 - 267 T^{2} + 32162 T^{4} - 2202431 T^{6} + 32162 p^{2} T^{8} - 267 p^{4} T^{10} + p^{6} T^{12} \) |
| 59 | \( ( 1 + 5 T + 182 T^{2} + 589 T^{3} + 182 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 61 | \( ( 1 + 20 T + 257 T^{2} + 2514 T^{3} + 257 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 67 | \( 1 - 207 T^{2} + 24158 T^{4} - 1953683 T^{6} + 24158 p^{2} T^{8} - 207 p^{4} T^{10} + p^{6} T^{12} \) |
| 71 | \( ( 1 + 7 T + 164 T^{2} + 989 T^{3} + 164 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 73 | \( 1 - 230 T^{2} + 29179 T^{4} - 2544392 T^{6} + 29179 p^{2} T^{8} - 230 p^{4} T^{10} + p^{6} T^{12} \) |
| 79 | \( ( 1 - 4 T + 109 T^{2} - 1048 T^{3} + 109 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( 1 - 283 T^{2} + 42918 T^{4} - 4366939 T^{6} + 42918 p^{2} T^{8} - 283 p^{4} T^{10} + p^{6} T^{12} \) |
| 89 | \( ( 1 + 2 T + 215 T^{2} + 316 T^{3} + 215 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 97 | \( 1 - 2 p T^{2} + 30991 T^{4} - 3405452 T^{6} + 30991 p^{2} T^{8} - 2 p^{5} T^{10} + p^{6} T^{12} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.01295195184554566015188705112, −4.87783537370614324363435415157, −4.71867043498078344506335484083, −4.60618732892329542147131398640, −4.48107189326942524583628264378, −4.06812720047121556237899066177, −4.05512352955260696291153529710, −3.97890346019327622627674007503, −3.90168852981352247993045995244, −3.39665812143977080970525631719, −3.35442041660878384463288588992, −3.25801566857281240583196657946, −3.06397088074925791257494830908, −2.83647520640854055661320221140, −2.77792864161789050613355794830, −2.58204324289950551159141520302, −2.16794299569408443005522095904, −2.11451482627319769464167510978, −2.08955760222196656314169816222, −1.41474282214450008622335414325, −1.40342519190880397215528845339, −1.27926848381581912832397892380, −1.05815017369385823794533804953, −0.45367968493702258984870384506, −0.099888995226492675145409627153,
0.099888995226492675145409627153, 0.45367968493702258984870384506, 1.05815017369385823794533804953, 1.27926848381581912832397892380, 1.40342519190880397215528845339, 1.41474282214450008622335414325, 2.08955760222196656314169816222, 2.11451482627319769464167510978, 2.16794299569408443005522095904, 2.58204324289950551159141520302, 2.77792864161789050613355794830, 2.83647520640854055661320221140, 3.06397088074925791257494830908, 3.25801566857281240583196657946, 3.35442041660878384463288588992, 3.39665812143977080970525631719, 3.90168852981352247993045995244, 3.97890346019327622627674007503, 4.05512352955260696291153529710, 4.06812720047121556237899066177, 4.48107189326942524583628264378, 4.60618732892329542147131398640, 4.71867043498078344506335484083, 4.87783537370614324363435415157, 5.01295195184554566015188705112
Plot not available for L-functions of degree greater than 10.
