| L(s) = 1 | − 2-s + 3-s − 4-s − 6-s − 4·7-s + 3·8-s + 9-s − 12-s + 4·14-s − 16-s − 18-s − 2·19-s − 4·21-s + 3·24-s + 6·25-s + 27-s + 4·28-s + 10·29-s − 5·32-s − 36-s + 2·38-s − 2·41-s + 4·42-s + 12·43-s − 48-s − 2·49-s − 6·50-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.408·6-s − 1.51·7-s + 1.06·8-s + 1/3·9-s − 0.288·12-s + 1.06·14-s − 1/4·16-s − 0.235·18-s − 0.458·19-s − 0.872·21-s + 0.612·24-s + 6/5·25-s + 0.192·27-s + 0.755·28-s + 1.85·29-s − 0.883·32-s − 1/6·36-s + 0.324·38-s − 0.312·41-s + 0.617·42-s + 1.82·43-s − 0.144·48-s − 2/7·49-s − 0.848·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 623808 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 623808 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9906473984\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9906473984\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 3 | $C_1$ | \( 1 - T \) |
| 19 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
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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
−8.608317699298466740616983204702, −8.059518497776761313911728381977, −7.60996337353470976276133542506, −6.97187978479941081941331789083, −6.81546890914088866573194892776, −6.21689903095898486627636167543, −5.79039434939135697204507886215, −4.93939794927448804758755856176, −4.66387955998219155801246154419, −4.01277093597698330553229736687, −3.52765775480504781810269151518, −2.88655456359875439506823633271, −2.48532838607163624830753689314, −1.41570887306683572230055281074, −0.58619109661222058739001863071,
0.58619109661222058739001863071, 1.41570887306683572230055281074, 2.48532838607163624830753689314, 2.88655456359875439506823633271, 3.52765775480504781810269151518, 4.01277093597698330553229736687, 4.66387955998219155801246154419, 4.93939794927448804758755856176, 5.79039434939135697204507886215, 6.21689903095898486627636167543, 6.81546890914088866573194892776, 6.97187978479941081941331789083, 7.60996337353470976276133542506, 8.059518497776761313911728381977, 8.608317699298466740616983204702
