| L(s) = 1 | − 2-s + 4·5-s + 2·7-s + 8-s − 4·10-s + 5·11-s + 7·13-s − 2·14-s − 16-s − 6·17-s − 5·22-s − 5·23-s + 2·25-s − 7·26-s + 4·29-s + 20·31-s + 6·34-s + 8·35-s − 6·37-s + 4·40-s + 2·43-s + 5·46-s + 26·47-s + 7·49-s − 2·50-s − 12·53-s + 20·55-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.78·5-s + 0.755·7-s + 0.353·8-s − 1.26·10-s + 1.50·11-s + 1.94·13-s − 0.534·14-s − 1/4·16-s − 1.45·17-s − 1.06·22-s − 1.04·23-s + 2/5·25-s − 1.37·26-s + 0.742·29-s + 3.59·31-s + 1.02·34-s + 1.35·35-s − 0.986·37-s + 0.632·40-s + 0.304·43-s + 0.737·46-s + 3.79·47-s + 49-s − 0.282·50-s − 1.64·53-s + 2.69·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 492804 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 492804 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.524897185\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.524897185\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 13 | $C_2$ | \( 1 - 7 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 2 T - 3 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 5 T + 14 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 5 T + 2 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 4 T - 13 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 6 T - T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 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 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 5 T - 34 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 10 T + 33 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 8 T - 7 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 17 T + 192 T^{2} + 17 p T^{3} + p^{2} T^{4} \) |
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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
−10.50034096822641886117538386922, −10.31980084018333179725570999432, −9.636891065324781429435712905438, −9.423882734394497125807968345295, −8.806292826762759857454750949693, −8.792763787445136309272789331210, −8.151556563677392041227260128995, −7.972069724836776738087836246446, −6.81627541764353897232085021577, −6.80869108916040031238838285987, −6.05185208606848493716219875607, −6.04861569875053879470978792493, −5.48508711354488382383005589466, −4.58476305409998666295127218709, −4.18538243305469637014629575394, −3.88631947451703367998879168970, −2.66180625516763470328315366953, −2.25387442779167571507744502693, −1.34177446234938717115764028993, −1.21476067643273336426326355457,
1.21476067643273336426326355457, 1.34177446234938717115764028993, 2.25387442779167571507744502693, 2.66180625516763470328315366953, 3.88631947451703367998879168970, 4.18538243305469637014629575394, 4.58476305409998666295127218709, 5.48508711354488382383005589466, 6.04861569875053879470978792493, 6.05185208606848493716219875607, 6.80869108916040031238838285987, 6.81627541764353897232085021577, 7.972069724836776738087836246446, 8.151556563677392041227260128995, 8.792763787445136309272789331210, 8.806292826762759857454750949693, 9.423882734394497125807968345295, 9.636891065324781429435712905438, 10.31980084018333179725570999432, 10.50034096822641886117538386922
