L(s) = 1 | − 2-s − 2·3-s + 4-s + 2·6-s + 2·7-s − 8-s + 9-s − 2·12-s − 2·14-s + 16-s + 12·17-s − 18-s − 4·21-s + 2·24-s − 5·25-s + 4·27-s + 2·28-s − 32-s − 12·34-s + 36-s + 4·42-s + 16·43-s − 2·48-s + 3·49-s + 5·50-s − 24·51-s + 12·53-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.15·3-s + 1/2·4-s + 0.816·6-s + 0.755·7-s − 0.353·8-s + 1/3·9-s − 0.577·12-s − 0.534·14-s + 1/4·16-s + 2.91·17-s − 0.235·18-s − 0.872·21-s + 0.408·24-s − 25-s + 0.769·27-s + 0.377·28-s − 0.176·32-s − 2.05·34-s + 1/6·36-s + 0.617·42-s + 2.43·43-s − 0.288·48-s + 3/7·49-s + 0.707·50-s − 3.36·51-s + 1.64·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 352800 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 352800 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.013615498\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.013615498\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 3 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 5 | $C_2$ | \( 1 + p T^{2} \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
good | 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−8.641946436905462506660121810666, −8.245441784546572320334189108752, −7.61188392109476428586562825239, −7.57571100088867902110310233811, −7.00040208558644603028855906365, −6.24370967191199643899749739066, −5.77713278642514260434446052217, −5.57928681742950427486583645839, −5.14472978179797247228927582624, −4.37172349798127682503408367548, −3.78880803467157430054818673540, −3.08226175311992974929178698307, −2.33138830690794607760289984480, −1.34951115258711429409173467390, −0.77732160932427153121684324644,
0.77732160932427153121684324644, 1.34951115258711429409173467390, 2.33138830690794607760289984480, 3.08226175311992974929178698307, 3.78880803467157430054818673540, 4.37172349798127682503408367548, 5.14472978179797247228927582624, 5.57928681742950427486583645839, 5.77713278642514260434446052217, 6.24370967191199643899749739066, 7.00040208558644603028855906365, 7.57571100088867902110310233811, 7.61188392109476428586562825239, 8.245441784546572320334189108752, 8.641946436905462506660121810666