| L(s) = 1 | − 5·3-s + 9·5-s + 28·7-s + 27·9-s − 57·11-s − 140·13-s − 45·15-s − 51·17-s + 5·19-s − 140·21-s + 69·23-s + 125·25-s − 280·27-s + 228·29-s + 23·31-s + 285·33-s + 252·35-s + 253·37-s + 700·39-s − 84·41-s + 248·43-s + 243·45-s + 201·47-s + 441·49-s + 255·51-s + 393·53-s − 513·55-s + ⋯ |
| L(s) = 1 | − 0.962·3-s + 0.804·5-s + 1.51·7-s + 9-s − 1.56·11-s − 2.98·13-s − 0.774·15-s − 0.727·17-s + 0.0603·19-s − 1.45·21-s + 0.625·23-s + 25-s − 1.99·27-s + 1.45·29-s + 0.133·31-s + 1.50·33-s + 1.21·35-s + 1.12·37-s + 2.87·39-s − 0.319·41-s + 0.879·43-s + 0.804·45-s + 0.623·47-s + 9/7·49-s + 0.700·51-s + 1.01·53-s − 1.25·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12544 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12544 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.456884237\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.456884237\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - 4 p T + p^{3} T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + 5 T - 2 T^{2} + 5 p^{3} T^{3} + p^{6} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 9 T - 44 T^{2} - 9 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 57 T + 1918 T^{2} + 57 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 70 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 3 p T - 8 p^{2} T^{2} + 3 p^{4} T^{3} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 5 T - 6834 T^{2} - 5 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 3 p T - 14 p^{2} T^{2} - 3 p^{4} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 114 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 23 T - 29262 T^{2} - 23 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 253 T + 13356 T^{2} - 253 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 42 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 124 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 201 T - 63422 T^{2} - 201 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 393 T + 5572 T^{2} - 393 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 219 T - 157418 T^{2} - 219 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 709 T + 275700 T^{2} - 709 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 419 T - 125202 T^{2} - 419 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 96 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 313 T - 291048 T^{2} - 313 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 461 T - 280518 T^{2} - 461 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 588 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 1017 T + 329320 T^{2} - 1017 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 1834 T + p^{3} T^{2} )^{2} \) |
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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
−13.60782332685305149632238179424, −12.66837681099206710461923172594, −12.38589878609169320169845936982, −11.94157378300821387265262199356, −11.12617753905737023704994500914, −10.92312515150809899999584688326, −10.24787097418557539767231598488, −9.822062218638266939600391947917, −9.408072128830228046476294858753, −8.371835607442038289036460901721, −7.86698863812020588520640571484, −7.20352261558888564560603235313, −6.92071386332276326703817062407, −5.78360694535227747061701620313, −5.16332039021654588844110979129, −4.93853429110145066101634325214, −4.40566887810134080486787917437, −2.43984266907745830284815425869, −2.27991706446225172238720019559, −0.68833255702630976640643116535,
0.68833255702630976640643116535, 2.27991706446225172238720019559, 2.43984266907745830284815425869, 4.40566887810134080486787917437, 4.93853429110145066101634325214, 5.16332039021654588844110979129, 5.78360694535227747061701620313, 6.92071386332276326703817062407, 7.20352261558888564560603235313, 7.86698863812020588520640571484, 8.371835607442038289036460901721, 9.408072128830228046476294858753, 9.822062218638266939600391947917, 10.24787097418557539767231598488, 10.92312515150809899999584688326, 11.12617753905737023704994500914, 11.94157378300821387265262199356, 12.38589878609169320169845936982, 12.66837681099206710461923172594, 13.60782332685305149632238179424
