| L(s) = 1 | − 2-s + 3-s + 4-s − 2·5-s − 6-s − 8-s + 9-s + 2·10-s + 12-s − 2·15-s + 16-s − 18-s − 8·19-s − 2·20-s − 24-s + 3·25-s + 27-s − 12·29-s + 2·30-s − 32-s + 36-s + 8·38-s + 2·40-s − 8·43-s − 2·45-s + 48-s + 2·49-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.894·5-s − 0.408·6-s − 0.353·8-s + 1/3·9-s + 0.632·10-s + 0.288·12-s − 0.516·15-s + 1/4·16-s − 0.235·18-s − 1.83·19-s − 0.447·20-s − 0.204·24-s + 3/5·25-s + 0.192·27-s − 2.22·29-s + 0.365·30-s − 0.176·32-s + 1/6·36-s + 1.29·38-s + 0.316·40-s − 1.21·43-s − 0.298·45-s + 0.144·48-s + 2/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 86400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 86400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 + T \) |
| 3 | $C_1$ | \( 1 - T \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−9.305587122869086271308905422277, −8.839725981578544270137072045579, −8.525273921993024774478731183093, −7.941231322257043910223245152113, −7.56448247571969758847240189448, −7.14136557129718457159189751168, −6.42217617652666865799421983967, −6.06876003878695035749777283902, −5.11622912879840160940122562021, −4.50780010715376054477637910369, −3.78892100904656137330232404529, −3.36585804145949552210873743484, −2.38545695127207581829171990462, −1.67076527604453903206835464150, 0,
1.67076527604453903206835464150, 2.38545695127207581829171990462, 3.36585804145949552210873743484, 3.78892100904656137330232404529, 4.50780010715376054477637910369, 5.11622912879840160940122562021, 6.06876003878695035749777283902, 6.42217617652666865799421983967, 7.14136557129718457159189751168, 7.56448247571969758847240189448, 7.941231322257043910223245152113, 8.525273921993024774478731183093, 8.839725981578544270137072045579, 9.305587122869086271308905422277
