| L(s) = 1 | + 3·2-s + 3·4-s − 4·7-s − 3·8-s + 7·11-s − 12·14-s − 13·16-s + 21·22-s + 7·23-s − 25-s − 12·28-s − 11·29-s − 15·32-s − 11·37-s − 11·43-s + 21·44-s + 21·46-s + 9·49-s − 3·50-s + 12·53-s + 12·56-s − 33·58-s + 3·64-s + 4·67-s − 14·71-s − 33·74-s − 28·77-s + ⋯ |
| L(s) = 1 | + 2.12·2-s + 3/2·4-s − 1.51·7-s − 1.06·8-s + 2.11·11-s − 3.20·14-s − 3.25·16-s + 4.47·22-s + 1.45·23-s − 1/5·25-s − 2.26·28-s − 2.04·29-s − 2.65·32-s − 1.80·37-s − 1.67·43-s + 3.16·44-s + 3.09·46-s + 9/7·49-s − 0.424·50-s + 1.64·53-s + 1.60·56-s − 4.33·58-s + 3/8·64-s + 0.488·67-s − 1.66·71-s − 3.83·74-s − 3.19·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 490931 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490931 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 12 T + p T^{2} ) \) |
| 233 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 17 T + p T^{2} ) \) |
| good | 2 | $C_2$$\times$$C_2$ | \( ( 1 - p T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 3 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 28 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 24 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 31 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 15 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 23 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + 23 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 160 T^{2} + p^{2} T^{4} \) |
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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
−8.664661374828560006517928432444, −7.51419878345776476895656049749, −6.90493869783676048795445523203, −6.81502152752977894984630037781, −6.31176939053061048064459795969, −5.83288254627329073984440093653, −5.44812737047945220470486651350, −4.97876014032945039057441134814, −4.34044636567938521802042748416, −3.81588306536316147667079427256, −3.55983058220673153140508470846, −3.27368896480593856424682989775, −2.50546353756040317442935144100, −1.44976440109687326095439055240, 0,
1.44976440109687326095439055240, 2.50546353756040317442935144100, 3.27368896480593856424682989775, 3.55983058220673153140508470846, 3.81588306536316147667079427256, 4.34044636567938521802042748416, 4.97876014032945039057441134814, 5.44812737047945220470486651350, 5.83288254627329073984440093653, 6.31176939053061048064459795969, 6.81502152752977894984630037781, 6.90493869783676048795445523203, 7.51419878345776476895656049749, 8.664661374828560006517928432444
