| L(s) = 1 | + 2-s − 3-s − 5-s − 6-s − 8-s − 10-s + 15-s − 16-s − 3·17-s − 3·19-s − 4·23-s + 24-s + 27-s + 30-s + 2·31-s − 3·34-s − 3·38-s + 40-s − 4·46-s + 48-s + 49-s + 3·51-s + 54-s + 3·57-s + 2·62-s + 64-s + 4·69-s + ⋯ |
| L(s) = 1 | + 2-s − 3-s − 5-s − 6-s − 8-s − 10-s + 15-s − 16-s − 3·17-s − 3·19-s − 4·23-s + 24-s + 27-s + 30-s + 2·31-s − 3·34-s − 3·38-s + 40-s − 4·46-s + 48-s + 49-s + 3·51-s + 54-s + 3·57-s + 2·62-s + 64-s + 4·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3459600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3459600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.02137688031\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.02137688031\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 3 | $C_2$ | \( 1 + T + T^{2} \) |
| 5 | $C_2$ | \( 1 + T + T^{2} \) |
| 31 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
| 23 | $C_1$ | \( ( 1 + T )^{4} \) |
| 29 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 71 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 73 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 83 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.10823562432668908806115536355, −9.074161476847201786665245082471, −8.556601155658246301973932538455, −8.533451082316832587064430392932, −8.208332825457758440542213104709, −7.74811431071225108468416519154, −6.94382061982305856191300411782, −6.59629879326233210355278359956, −6.30747225930340887947205421279, −6.09107812417567590873243082713, −5.77267713192476348700344454791, −5.01158946893955614561058821181, −4.42801517784651418863427104037, −4.38794171526591026738619477018, −4.00666711565430950102062583216, −3.85192981170327075673865920864, −2.61006877595871341558296629138, −2.44701808039901576161071865951, −1.89474527404143668492961564638, −0.090280572630585089917016260268,
0.090280572630585089917016260268, 1.89474527404143668492961564638, 2.44701808039901576161071865951, 2.61006877595871341558296629138, 3.85192981170327075673865920864, 4.00666711565430950102062583216, 4.38794171526591026738619477018, 4.42801517784651418863427104037, 5.01158946893955614561058821181, 5.77267713192476348700344454791, 6.09107812417567590873243082713, 6.30747225930340887947205421279, 6.59629879326233210355278359956, 6.94382061982305856191300411782, 7.74811431071225108468416519154, 8.208332825457758440542213104709, 8.533451082316832587064430392932, 8.556601155658246301973932538455, 9.074161476847201786665245082471, 10.10823562432668908806115536355
