| L(s) = 1 | − 2·2-s − 2·3-s + 2·4-s − 4·5-s + 4·6-s − 4·7-s + 2·9-s + 8·10-s − 4·12-s + 8·14-s + 8·15-s − 4·16-s + 10·17-s − 4·18-s − 2·19-s − 8·20-s + 8·21-s + 8·23-s + 11·25-s − 6·27-s − 8·28-s − 16·30-s + 8·32-s − 20·34-s + 16·35-s + 4·36-s + 4·38-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.15·3-s + 4-s − 1.78·5-s + 1.63·6-s − 1.51·7-s + 2/3·9-s + 2.52·10-s − 1.15·12-s + 2.13·14-s + 2.06·15-s − 16-s + 2.42·17-s − 0.942·18-s − 0.458·19-s − 1.78·20-s + 1.74·21-s + 1.66·23-s + 11/5·25-s − 1.15·27-s − 1.51·28-s − 2.92·30-s + 1.41·32-s − 3.42·34-s + 2.70·35-s + 2/3·36-s + 0.648·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 144400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1664762109\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1664762109\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 5 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| 19 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 12 T + 72 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 20 T + 200 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 20 T + 200 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
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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
−11.63208236816003624210742197675, −11.15639221790371607001138036526, −10.66514855508434260980175923208, −10.11634113022597540682281333749, −9.803573576984259043241855171986, −9.494366261446820918221325504792, −8.766324518902652080421493177813, −8.340766529103677760043828393316, −7.81273941244522282525299847116, −7.49058586675796400849748712935, −6.91556295480652558461808133889, −6.64130475966195335784825733303, −6.01989445821401619339399391226, −5.19787086294410104352020598983, −4.88586476572418511361912691176, −3.85947417847140959498101092628, −3.49799430073330169701001365647, −2.85659164645844135043673340268, −1.25119970149498671791291730699, −0.42470211655815317100259624463,
0.42470211655815317100259624463, 1.25119970149498671791291730699, 2.85659164645844135043673340268, 3.49799430073330169701001365647, 3.85947417847140959498101092628, 4.88586476572418511361912691176, 5.19787086294410104352020598983, 6.01989445821401619339399391226, 6.64130475966195335784825733303, 6.91556295480652558461808133889, 7.49058586675796400849748712935, 7.81273941244522282525299847116, 8.340766529103677760043828393316, 8.766324518902652080421493177813, 9.494366261446820918221325504792, 9.803573576984259043241855171986, 10.11634113022597540682281333749, 10.66514855508434260980175923208, 11.15639221790371607001138036526, 11.63208236816003624210742197675
