| L(s) = 1 | + 5·5-s − 32·7-s + 36·11-s + 10·13-s + 156·17-s + 280·19-s − 192·23-s + 6·29-s + 16·31-s − 160·35-s − 68·37-s − 390·41-s + 52·43-s + 408·47-s + 343·49-s + 228·53-s + 180·55-s + 516·59-s + 58·61-s + 50·65-s + 892·67-s + 240·71-s − 1.29e3·73-s − 1.15e3·77-s + 1.16e3·79-s − 732·83-s + 780·85-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.72·7-s + 0.986·11-s + 0.213·13-s + 2.22·17-s + 3.38·19-s − 1.74·23-s + 0.0384·29-s + 0.0926·31-s − 0.772·35-s − 0.302·37-s − 1.48·41-s + 0.184·43-s + 1.26·47-s + 49-s + 0.590·53-s + 0.441·55-s + 1.13·59-s + 0.121·61-s + 0.0954·65-s + 1.62·67-s + 0.401·71-s − 2.07·73-s − 1.70·77-s + 1.66·79-s − 0.968·83-s + 0.995·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2624400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2624400 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.815352182\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.815352182\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - p T + p^{2} T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 32 T + 681 T^{2} + 32 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 36 T - 35 T^{2} - 36 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 10 T - 2097 T^{2} - 10 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 78 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 140 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 192 T + 24697 T^{2} + 192 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 6 T - 24353 T^{2} - 6 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 16 T - 29535 T^{2} - 16 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 34 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 390 T + 83179 T^{2} + 390 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 52 T - 76803 T^{2} - 52 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 408 T + 62641 T^{2} - 408 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 114 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 516 T + 60877 T^{2} - 516 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 58 T - 223617 T^{2} - 58 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 892 T + 494901 T^{2} - 892 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 120 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 646 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 1168 T + 871185 T^{2} - 1168 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 732 T - 35963 T^{2} + 732 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 1590 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 2 p T - 93 p^{2} T^{2} + 2 p^{4} T^{3} + p^{6} T^{4} \) |
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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
−9.355700014624795915165473210464, −8.957443219870005762100105087169, −8.559191282727349600772239756330, −7.87363911714823365878502758221, −7.50061521551121798022346313356, −7.41502357686374101558462391567, −6.66372111873311704582122268005, −6.49982454698800300044380698471, −5.80661186852532379381577915685, −5.73585567407966762357358433129, −5.30347347061979727225001769932, −4.81702235819604419058825297831, −3.78206363679394617051609795565, −3.73188987453581386795549364296, −3.23082141514231525811268273839, −3.04872974283581631192514055274, −2.16088284888887560299574468514, −1.54381115221371941539635474003, −0.78532214588856797542208634786, −0.69545023403847283485533183278,
0.69545023403847283485533183278, 0.78532214588856797542208634786, 1.54381115221371941539635474003, 2.16088284888887560299574468514, 3.04872974283581631192514055274, 3.23082141514231525811268273839, 3.73188987453581386795549364296, 3.78206363679394617051609795565, 4.81702235819604419058825297831, 5.30347347061979727225001769932, 5.73585567407966762357358433129, 5.80661186852532379381577915685, 6.49982454698800300044380698471, 6.66372111873311704582122268005, 7.41502357686374101558462391567, 7.50061521551121798022346313356, 7.87363911714823365878502758221, 8.559191282727349600772239756330, 8.957443219870005762100105087169, 9.355700014624795915165473210464
