| L(s) = 1 | + 2·2-s + 3-s + 3·4-s + 2·6-s − 7-s + 4·8-s + 9-s + 3·12-s − 2·14-s + 5·16-s + 2·18-s + 19-s − 21-s + 4·24-s − 25-s + 27-s − 3·28-s + 3·29-s + 6·32-s + 3·36-s + 2·38-s + 7·41-s − 2·42-s + 21·43-s + 5·48-s + 7·49-s − 2·50-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 0.577·3-s + 3/2·4-s + 0.816·6-s − 0.377·7-s + 1.41·8-s + 1/3·9-s + 0.866·12-s − 0.534·14-s + 5/4·16-s + 0.471·18-s + 0.229·19-s − 0.218·21-s + 0.816·24-s − 1/5·25-s + 0.192·27-s − 0.566·28-s + 0.557·29-s + 1.06·32-s + 1/2·36-s + 0.324·38-s + 1.09·41-s − 0.308·42-s + 3.20·43-s + 0.721·48-s + 49-s − 0.282·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 740772 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 740772 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.487178073\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.487178073\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 19 | $C_1$ | \( 1 - T \) |
| good | 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 24 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 37 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 9 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + 5 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + 7 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 37 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
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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
−8.029919450098150731601704226531, −7.79350713214530279059497827736, −7.21139404743658688986920902358, −7.03971304513101435193144304872, −6.31455272288020134183488031652, −5.89245710617586384411113709112, −5.69803593509908875286289864047, −4.96353323084480380892490147451, −4.34859427363888342632457965583, −4.19881128608252065643686481123, −3.56480018610139750566973789577, −2.86932136951985524784196277472, −2.68181791467429682872539478690, −1.91231073341082219916111392497, −1.00522315799540801804239512229,
1.00522315799540801804239512229, 1.91231073341082219916111392497, 2.68181791467429682872539478690, 2.86932136951985524784196277472, 3.56480018610139750566973789577, 4.19881128608252065643686481123, 4.34859427363888342632457965583, 4.96353323084480380892490147451, 5.69803593509908875286289864047, 5.89245710617586384411113709112, 6.31455272288020134183488031652, 7.03971304513101435193144304872, 7.21139404743658688986920902358, 7.79350713214530279059497827736, 8.029919450098150731601704226531
