| L(s) = 1 | − 2-s + 2·7-s + 8-s − 2·9-s − 2·14-s − 16-s + 2·18-s + 2·23-s − 2·46-s + 3·49-s + 2·56-s − 4·63-s + 64-s + 2·71-s − 2·72-s + 2·79-s + 3·81-s − 3·98-s − 2·112-s − 2·113-s − 121-s + 4·126-s + 127-s − 128-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 2-s + 2·7-s + 8-s − 2·9-s − 2·14-s − 16-s + 2·18-s + 2·23-s − 2·46-s + 3·49-s + 2·56-s − 4·63-s + 64-s + 2·71-s − 2·72-s + 2·79-s + 3·81-s − 3·98-s − 2·112-s − 2·113-s − 121-s + 4·126-s + 127-s − 128-s + 131-s + 137-s + 139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6676389305\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6676389305\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 19 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 59 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
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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
−9.779024569338101610645830766425, −9.420855792220089780829821102286, −8.974190698158375424781899021951, −8.752864014611837125206330724510, −8.374357162294601641240779888785, −8.175960017039092695013840151809, −7.59126366490503593951134228904, −7.51506998379007194065014130210, −6.80216194492556374663850481879, −6.36151742882763868532942070243, −5.74418343413207275652897321280, −5.14470689008525013489662826487, −5.06511350778806645121820480257, −4.77125087247963427399857683850, −3.90274245315059242989351925157, −3.52479085271814207905637985186, −2.53036363345507421042197738697, −2.43350354613194614869655619725, −1.49688086274837542164910117526, −0.890299391740152946017137073877,
0.890299391740152946017137073877, 1.49688086274837542164910117526, 2.43350354613194614869655619725, 2.53036363345507421042197738697, 3.52479085271814207905637985186, 3.90274245315059242989351925157, 4.77125087247963427399857683850, 5.06511350778806645121820480257, 5.14470689008525013489662826487, 5.74418343413207275652897321280, 6.36151742882763868532942070243, 6.80216194492556374663850481879, 7.51506998379007194065014130210, 7.59126366490503593951134228904, 8.175960017039092695013840151809, 8.374357162294601641240779888785, 8.752864014611837125206330724510, 8.974190698158375424781899021951, 9.420855792220089780829821102286, 9.779024569338101610645830766425
