| L(s) = 1 | − 2-s − 4·3-s + 4-s + 4·6-s − 8-s + 6·9-s − 4·12-s + 16-s + 12·17-s − 6·18-s + 4·19-s + 4·24-s − 10·25-s + 4·27-s − 32-s − 12·34-s + 6·36-s − 4·38-s + 12·41-s + 16·43-s − 4·48-s + 49-s + 10·50-s − 48·51-s − 4·54-s − 16·57-s − 12·59-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 2.30·3-s + 1/2·4-s + 1.63·6-s − 0.353·8-s + 2·9-s − 1.15·12-s + 1/4·16-s + 2.91·17-s − 1.41·18-s + 0.917·19-s + 0.816·24-s − 2·25-s + 0.769·27-s − 0.176·32-s − 2.05·34-s + 36-s − 0.648·38-s + 1.87·41-s + 2.43·43-s − 0.577·48-s + 1/7·49-s + 1.41·50-s − 6.72·51-s − 0.544·54-s − 2.11·57-s − 1.56·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6272 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6272 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3095066963\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3095066963\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 3 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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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
−12.03047354988920041272999065688, −11.23136141438460806904855801814, −11.08163105431410064794151744331, −10.39623059414136659793705857031, −9.765547119459919407856461234632, −9.479912708108790860766004869191, −8.328602349022309226048251750698, −7.57571100088867902110310233811, −7.29200576171944643329126204500, −6.04992362572003306544038558924, −5.73625972355331222443538857924, −5.57928681742950427486583645839, −4.39126934619586232082133319537, −3.09628357761679869111062402047, −1.00403210582709168291350506092,
1.00403210582709168291350506092, 3.09628357761679869111062402047, 4.39126934619586232082133319537, 5.57928681742950427486583645839, 5.73625972355331222443538857924, 6.04992362572003306544038558924, 7.29200576171944643329126204500, 7.57571100088867902110310233811, 8.328602349022309226048251750698, 9.479912708108790860766004869191, 9.765547119459919407856461234632, 10.39623059414136659793705857031, 11.08163105431410064794151744331, 11.23136141438460806904855801814, 12.03047354988920041272999065688
