| L(s) = 1 | + 2-s − 3·5-s + 5·7-s − 8-s − 3·10-s − 3·11-s + 10·13-s + 5·14-s − 16-s + 3·17-s − 5·19-s − 3·22-s − 3·23-s + 5·25-s + 10·26-s + 6·29-s + 4·31-s + 3·34-s − 15·35-s + 7·37-s − 5·38-s + 3·40-s + 18·41-s + 22·43-s − 3·46-s + 18·49-s + 5·50-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.34·5-s + 1.88·7-s − 0.353·8-s − 0.948·10-s − 0.904·11-s + 2.77·13-s + 1.33·14-s − 1/4·16-s + 0.727·17-s − 1.14·19-s − 0.639·22-s − 0.625·23-s + 25-s + 1.96·26-s + 1.11·29-s + 0.718·31-s + 0.514·34-s − 2.53·35-s + 1.15·37-s − 0.811·38-s + 0.474·40-s + 2.81·41-s + 3.35·43-s − 0.442·46-s + 18/7·49-s + 0.707·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.588272311\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.588272311\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - 5 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 5 T + 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 3 T - 14 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 7 T + 12 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 3 T - 44 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 12 T + 85 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 2 T - 57 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 11 T + 48 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 15 T + 136 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
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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
−10.34220122028087596798656835515, −9.522504121358386028611687476099, −8.891226833385801878122406863794, −8.722733110802035391253650484361, −8.167158492048686555481550602543, −8.101876096637942012191884497065, −7.56458326776227314569478269088, −7.48532144492719564883978653468, −6.40692489972887638870109508637, −6.22620596868304138213600402021, −5.64218873523986947280019276118, −5.42642939739238448349896038694, −4.59042064991422698850656927457, −4.18169729592797655969431318762, −4.15045448100220269358907368681, −3.69469586173892563049173973377, −2.61358185863628775441037354243, −2.55290428672454911294736394191, −1.21312791672291179919044185279, −0.931380735495143725136401995693,
0.931380735495143725136401995693, 1.21312791672291179919044185279, 2.55290428672454911294736394191, 2.61358185863628775441037354243, 3.69469586173892563049173973377, 4.15045448100220269358907368681, 4.18169729592797655969431318762, 4.59042064991422698850656927457, 5.42642939739238448349896038694, 5.64218873523986947280019276118, 6.22620596868304138213600402021, 6.40692489972887638870109508637, 7.48532144492719564883978653468, 7.56458326776227314569478269088, 8.101876096637942012191884497065, 8.167158492048686555481550602543, 8.722733110802035391253650484361, 8.891226833385801878122406863794, 9.522504121358386028611687476099, 10.34220122028087596798656835515
