| L(s) = 1 | − 2-s − 3·3-s − 4-s + 3·6-s + 3·8-s + 6·9-s + 3·12-s − 6·13-s − 16-s − 6·18-s + 14·23-s − 9·24-s − 10·25-s + 6·26-s − 9·27-s − 5·32-s − 6·36-s + 10·37-s + 18·39-s − 14·46-s − 4·47-s + 3·48-s + 2·49-s + 10·50-s + 6·52-s + 9·54-s − 4·59-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.73·3-s − 1/2·4-s + 1.22·6-s + 1.06·8-s + 2·9-s + 0.866·12-s − 1.66·13-s − 1/4·16-s − 1.41·18-s + 2.91·23-s − 1.83·24-s − 2·25-s + 1.17·26-s − 1.73·27-s − 0.883·32-s − 36-s + 1.64·37-s + 2.88·39-s − 2.06·46-s − 0.583·47-s + 0.433·48-s + 2/7·49-s + 1.41·50-s + 0.832·52-s + 1.22·54-s − 0.520·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 404496 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 404496 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 3 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
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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
−8.438091143870918709556369637413, −7.71828535390149419647213811344, −7.46412153801181724464039516931, −7.20736947999270954805514319970, −6.56076063238929550237884032031, −6.04347889941024738587947019148, −5.54011750528312868795733130111, −5.05905335905163628428589247731, −4.53638340152257253562350404113, −4.50862835068296220400083952922, −3.49318767761214532144325637053, −2.66263007986619920281397076025, −1.70090856454369339880391339176, −0.877937177229605012568584593461, 0,
0.877937177229605012568584593461, 1.70090856454369339880391339176, 2.66263007986619920281397076025, 3.49318767761214532144325637053, 4.50862835068296220400083952922, 4.53638340152257253562350404113, 5.05905335905163628428589247731, 5.54011750528312868795733130111, 6.04347889941024738587947019148, 6.56076063238929550237884032031, 7.20736947999270954805514319970, 7.46412153801181724464039516931, 7.71828535390149419647213811344, 8.438091143870918709556369637413
