L(s) = 1 | − 2·2-s + 3·4-s + 2·7-s − 4·8-s + 12·11-s − 4·14-s + 5·16-s − 24·22-s + 25-s + 6·28-s − 12·29-s − 6·32-s + 16·37-s + 16·43-s + 36·44-s − 3·49-s − 2·50-s − 12·53-s − 8·56-s + 24·58-s + 7·64-s − 8·67-s − 24·71-s − 32·74-s + 24·77-s − 8·79-s − 32·86-s + ⋯ |
L(s) = 1 | − 1.41·2-s + 3/2·4-s + 0.755·7-s − 1.41·8-s + 3.61·11-s − 1.06·14-s + 5/4·16-s − 5.11·22-s + 1/5·25-s + 1.13·28-s − 2.22·29-s − 1.06·32-s + 2.63·37-s + 2.43·43-s + 5.42·44-s − 3/7·49-s − 0.282·50-s − 1.64·53-s − 1.06·56-s + 3.15·58-s + 7/8·64-s − 0.977·67-s − 2.84·71-s − 3.71·74-s + 2.73·77-s − 0.900·79-s − 3.45·86-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 396900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 396900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.421127313\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.421127313\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
good | 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−8.808531037524113106929321069025, −8.361683788765927099886703120446, −7.68072566687187284194521624333, −7.33859557219653016096826874551, −7.10097996037311265247610233993, −6.21060412222094990512561243657, −6.15095560601501333547012713952, −5.76306799226879953191493212190, −4.42900889702114434346086151421, −4.39745787492882284514235740312, −3.67917655074737520443823096900, −3.02077554855234761959138967953, −1.98468742030680472449906296868, −1.50258067725827938602043215208, −0.945958207187822378507835712582,
0.945958207187822378507835712582, 1.50258067725827938602043215208, 1.98468742030680472449906296868, 3.02077554855234761959138967953, 3.67917655074737520443823096900, 4.39745787492882284514235740312, 4.42900889702114434346086151421, 5.76306799226879953191493212190, 6.15095560601501333547012713952, 6.21060412222094990512561243657, 7.10097996037311265247610233993, 7.33859557219653016096826874551, 7.68072566687187284194521624333, 8.361683788765927099886703120446, 8.808531037524113106929321069025