| L(s) = 1 | + 2-s − 2·3-s − 4-s + 2·5-s − 2·6-s − 2·7-s − 8-s + 9-s + 2·10-s + 2·12-s − 2·14-s − 4·15-s + 3·16-s + 18-s − 2·20-s + 4·21-s + 2·24-s + 2·25-s + 4·27-s + 2·28-s − 4·30-s + 31-s + 3·32-s − 4·35-s − 36-s − 5·37-s − 2·40-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.15·3-s − 1/2·4-s + 0.894·5-s − 0.816·6-s − 0.755·7-s − 0.353·8-s + 1/3·9-s + 0.632·10-s + 0.577·12-s − 0.534·14-s − 1.03·15-s + 3/4·16-s + 0.235·18-s − 0.447·20-s + 0.872·21-s + 0.408·24-s + 2/5·25-s + 0.769·27-s + 0.377·28-s − 0.730·30-s + 0.179·31-s + 0.530·32-s − 0.676·35-s − 1/6·36-s − 0.821·37-s − 0.316·40-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 620010 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 620010 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + p T^{2} ) \) |
| 3 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 3 T + p T^{2} ) \) |
| 83 | $C_2$ | \( 1 - 6 T + p T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 17 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 55 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 13 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 80 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 89 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + 52 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 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
−8.138779755064487795653155402102, −7.67477259915541951845603834545, −6.95925287780570246368006063301, −6.65585305378170051927394906979, −6.13481230182400022785156962937, −5.81258268923605130266867420375, −5.47585489403606264689783480067, −4.85259121433882753040455309052, −4.68890065441840651835695562382, −3.94634576270507223368910503550, −3.33005618474388535729400831605, −2.87781443628152275427991879285, −1.95443352918003550952555708332, −1.09526985189453765568977321512, 0,
1.09526985189453765568977321512, 1.95443352918003550952555708332, 2.87781443628152275427991879285, 3.33005618474388535729400831605, 3.94634576270507223368910503550, 4.68890065441840651835695562382, 4.85259121433882753040455309052, 5.47585489403606264689783480067, 5.81258268923605130266867420375, 6.13481230182400022785156962937, 6.65585305378170051927394906979, 6.95925287780570246368006063301, 7.67477259915541951845603834545, 8.138779755064487795653155402102
