| L(s) = 1 | + 8·3-s + 4·4-s + 30·9-s + 32·12-s + 12·13-s + 12·16-s − 96·25-s + 40·27-s + 120·36-s + 96·39-s + 160·43-s + 96·48-s + 176·49-s + 48·52-s − 160·61-s + 32·64-s − 768·75-s − 205·81-s − 384·100-s − 400·103-s + 160·108-s + 360·117-s − 340·121-s + 127-s + 1.28e3·129-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 8/3·3-s + 4-s + 10/3·9-s + 8/3·12-s + 0.923·13-s + 3/4·16-s − 3.83·25-s + 1.48·27-s + 10/3·36-s + 2.46·39-s + 3.72·43-s + 2·48-s + 3.59·49-s + 0.923·52-s − 2.62·61-s + 1/2·64-s − 10.2·75-s − 2.53·81-s − 3.83·100-s − 3.88·103-s + 1.48·108-s + 3.07·117-s − 2.80·121-s + 0.00787·127-s + 9.92·129-s + 0.00763·131-s + 0.00729·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37015056 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37015056 ^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(5.840012772\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.840012772\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 - 4 T + p^{2} T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p^{2} T^{2} )^{2} \) |
| good | 5 | $C_2^2$ | \( ( 1 + 48 T^{2} + p^{4} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 - 88 T^{2} + p^{4} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 170 T^{2} + p^{4} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 498 T^{2} + p^{4} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 632 T^{2} + p^{4} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 558 T^{2} + p^{4} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 1362 T^{2} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 968 T^{2} + p^{4} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 2578 T^{2} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + 1680 T^{2} + p^{4} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 40 T + p^{2} T^{2} )^{4} \) |
| 47 | $C_2^2$ | \( ( 1 + 890 T^{2} + p^{4} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 862 T^{2} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 4650 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 40 T + p^{2} T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 + 3272 T^{2} + p^{4} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 1370 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 418 T^{2} + p^{4} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + 330 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 6320 T^{2} + p^{4} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 + 4222 T^{2} + p^{4} T^{4} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.52155572004759870464728971837, −10.12192746372902868095240438531, −9.482828730616025937655793972708, −9.469774639251885134484756243795, −9.378310960386705031135875087726, −9.097205124156576141915710417450, −8.464354830430694199189827086137, −8.283763130573230732553394628403, −8.176203355931043482603169376033, −7.73183499799736860383954283400, −7.42234055790433553885394690271, −7.26884837763787259420116234159, −7.07842816559772822445943207988, −6.13983223496212790061968875805, −6.02451983432616560817872377844, −5.73175319232383440422061143078, −5.51499024942718477071263594127, −4.25776239785669113715597325547, −4.21075385454019022819401152652, −3.85729600371725446777376912338, −3.43302904989104165130684399493, −2.87235727200968367749786338101, −2.35265777731350037673982220974, −2.21730881694109526366018921046, −1.45199268545537457861500107218,
1.45199268545537457861500107218, 2.21730881694109526366018921046, 2.35265777731350037673982220974, 2.87235727200968367749786338101, 3.43302904989104165130684399493, 3.85729600371725446777376912338, 4.21075385454019022819401152652, 4.25776239785669113715597325547, 5.51499024942718477071263594127, 5.73175319232383440422061143078, 6.02451983432616560817872377844, 6.13983223496212790061968875805, 7.07842816559772822445943207988, 7.26884837763787259420116234159, 7.42234055790433553885394690271, 7.73183499799736860383954283400, 8.176203355931043482603169376033, 8.283763130573230732553394628403, 8.464354830430694199189827086137, 9.097205124156576141915710417450, 9.378310960386705031135875087726, 9.469774639251885134484756243795, 9.482828730616025937655793972708, 10.12192746372902868095240438531, 10.52155572004759870464728971837
