| L(s) = 1 | + 2-s − 3·3-s + 6·5-s − 3·6-s − 8-s + 6·9-s + 6·10-s − 12·11-s + 2·13-s − 18·15-s − 16-s + 6·17-s + 6·18-s − 7·19-s − 12·22-s + 6·23-s + 3·24-s + 17·25-s + 2·26-s − 9·27-s − 6·29-s − 18·30-s + 2·31-s + 36·33-s + 6·34-s − 2·37-s − 7·38-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.73·3-s + 2.68·5-s − 1.22·6-s − 0.353·8-s + 2·9-s + 1.89·10-s − 3.61·11-s + 0.554·13-s − 4.64·15-s − 1/4·16-s + 1.45·17-s + 1.41·18-s − 1.60·19-s − 2.55·22-s + 1.25·23-s + 0.612·24-s + 17/5·25-s + 0.392·26-s − 1.73·27-s − 1.11·29-s − 3.28·30-s + 0.359·31-s + 6.26·33-s + 1.02·34-s − 0.328·37-s − 1.13·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.683159637\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.683159637\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 6 T + 7 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 2 T - 27 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 2 T - 39 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 5 T - 36 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 8 T - 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 2 T - 69 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 5 T - 54 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 12 T + 61 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 2 T - 93 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
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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
−10.31106610192273497233875645764, −10.04960348701013356482764538884, −9.954866412436909415002663225904, −9.267888026859682434022268407582, −8.815459493188615872713192038725, −7.993812401748534362187545478256, −7.905331962977485482298243145082, −7.03071482557490676891910841279, −6.75152689464507612793678363666, −6.01299632460171902322251540505, −5.79798236279229627477501253458, −5.60787513117802202013116452335, −5.30210322774295502559022299055, −4.82465305401791288204313968320, −4.57403210989057571317061840957, −3.35744777979472771512928002963, −2.85357183787191991023970093328, −2.14055827212391412098138252981, −1.79095417998116721247102282223, −0.59655324307133357072090456706,
0.59655324307133357072090456706, 1.79095417998116721247102282223, 2.14055827212391412098138252981, 2.85357183787191991023970093328, 3.35744777979472771512928002963, 4.57403210989057571317061840957, 4.82465305401791288204313968320, 5.30210322774295502559022299055, 5.60787513117802202013116452335, 5.79798236279229627477501253458, 6.01299632460171902322251540505, 6.75152689464507612793678363666, 7.03071482557490676891910841279, 7.905331962977485482298243145082, 7.993812401748534362187545478256, 8.815459493188615872713192038725, 9.267888026859682434022268407582, 9.954866412436909415002663225904, 10.04960348701013356482764538884, 10.31106610192273497233875645764
