| L(s) = 1 | + 128·4-s + 250·5-s + 962·9-s + 1.22e4·16-s − 1.44e4·19-s + 3.20e4·20-s + 4.68e4·25-s − 5.95e4·31-s + 1.23e5·36-s − 4.17e4·41-s + 2.40e5·45-s + 2.35e5·49-s + 6.71e5·59-s + 1.04e6·64-s − 9.99e5·71-s − 1.85e6·76-s + 3.07e6·80-s + 3.94e5·81-s − 3.61e6·95-s + 6.00e6·100-s + 3.79e6·101-s − 2.52e5·109-s + 3.54e6·121-s − 7.62e6·124-s + 7.81e6·125-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 2·4-s + 2·5-s + 1.31·9-s + 3·16-s − 2.11·19-s + 4·20-s + 3·25-s − 2·31-s + 2.63·36-s − 0.605·41-s + 2.63·45-s + 2·49-s + 3.26·59-s + 4·64-s − 2.79·71-s − 4.22·76-s + 6·80-s + 0.741·81-s − 4.22·95-s + 6·100-s + 3.68·101-s − 0.194·109-s + 2·121-s − 4·124-s + 4·125-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24025 ^{s/2} \, \Gamma_{\C}(s+3)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(10.28631135\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.28631135\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_1$ | \( ( 1 - p^{3} T )^{2} \) |
| 31 | $C_1$ | \( ( 1 + p^{3} T )^{2} \) |
| good | 2 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 3 | $C_2^2$ | \( 1 - 962 T^{2} + p^{12} T^{4} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 9593602 T^{2} + p^{12} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 26609362 T^{2} + p^{12} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 7238 T + p^{6} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 294307598 T^{2} + p^{12} T^{4} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 4616425762 T^{2} + p^{12} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 20878 T + p^{6} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 543440482 T^{2} + p^{12} T^{4} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 41167115678 T^{2} + p^{12} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 335738 T + p^{6} T^{2} )^{2} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 499502 T + p^{6} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 45257367922 T^{2} + p^{12} T^{4} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 419471864318 T^{2} + p^{12} T^{4} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
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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
−12.17225396817684494647850126586, −11.44428468273456348835125571071, −10.96669832054880885800797132317, −10.37687659554399380886027284909, −10.22694899202590980336476885460, −9.943423512812749039684511635204, −8.873195458393930170496933281734, −8.730177525992499571452554156572, −7.60991334980586652931059574678, −7.02201562030234661398256583244, −6.88918187293540616410750543105, −6.17425073017937910293039254661, −5.82809849043478597742559826954, −5.25407287703138698786970760506, −4.26034270547071415364284765674, −3.40873456625043113756187180587, −2.41616912218109480702524432224, −2.06953227437371208495552350108, −1.70174646743150872274300859649, −0.924886137248882523856428439921,
0.924886137248882523856428439921, 1.70174646743150872274300859649, 2.06953227437371208495552350108, 2.41616912218109480702524432224, 3.40873456625043113756187180587, 4.26034270547071415364284765674, 5.25407287703138698786970760506, 5.82809849043478597742559826954, 6.17425073017937910293039254661, 6.88918187293540616410750543105, 7.02201562030234661398256583244, 7.60991334980586652931059574678, 8.730177525992499571452554156572, 8.873195458393930170496933281734, 9.943423512812749039684511635204, 10.22694899202590980336476885460, 10.37687659554399380886027284909, 10.96669832054880885800797132317, 11.44428468273456348835125571071, 12.17225396817684494647850126586
