| L(s) = 1 | + 2·2-s + 3·4-s − 3·7-s + 4·8-s + 3·9-s − 6·14-s + 5·16-s + 6·18-s − 14·23-s + 6·25-s − 9·28-s − 16·29-s + 6·32-s + 9·36-s + 8·37-s − 10·43-s − 28·46-s + 2·49-s + 12·50-s − 12·53-s − 12·56-s − 32·58-s − 9·63-s + 7·64-s − 8·67-s + 2·71-s + 12·72-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s − 1.13·7-s + 1.41·8-s + 9-s − 1.60·14-s + 5/4·16-s + 1.41·18-s − 2.91·23-s + 6/5·25-s − 1.70·28-s − 2.97·29-s + 1.06·32-s + 3/2·36-s + 1.31·37-s − 1.52·43-s − 4.12·46-s + 2/7·49-s + 1.69·50-s − 1.64·53-s − 1.60·56-s − 4.20·58-s − 1.13·63-s + 7/8·64-s − 0.977·67-s + 0.237·71-s + 1.41·72-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 988036 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 988036 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_2$ | \( 1 + 3 T + p T^{2} \) |
| 71 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
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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
−7.74164320100167851666984821721, −7.43664690273418529041782556592, −6.79665889988949967719732090774, −6.49242262547273174623396635775, −6.06956565112627680353985311275, −5.75487565821591001758107260532, −5.18991100616205512936411884481, −4.59340547650806364343033240582, −4.17732178013697623495615501189, −3.69652118358615221384815805348, −3.41755261781539893527446126592, −2.71244755097108082099509431909, −1.99211087757081249885574149979, −1.55154160443356172633551919559, 0,
1.55154160443356172633551919559, 1.99211087757081249885574149979, 2.71244755097108082099509431909, 3.41755261781539893527446126592, 3.69652118358615221384815805348, 4.17732178013697623495615501189, 4.59340547650806364343033240582, 5.18991100616205512936411884481, 5.75487565821591001758107260532, 6.06956565112627680353985311275, 6.49242262547273174623396635775, 6.79665889988949967719732090774, 7.43664690273418529041782556592, 7.74164320100167851666984821721
