L(s) = 1 | + 3-s − 2·4-s − 7-s + 9-s + 11-s − 2·12-s + 4·16-s − 3·17-s − 3·19-s − 21-s + 23-s + 27-s + 2·28-s − 7·29-s + 6·31-s + 33-s − 2·36-s + 8·37-s − 2·41-s + 5·43-s − 2·44-s + 6·47-s + 4·48-s + 49-s − 3·51-s + 3·53-s − 3·57-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 4-s − 0.377·7-s + 1/3·9-s + 0.301·11-s − 0.577·12-s + 16-s − 0.727·17-s − 0.688·19-s − 0.218·21-s + 0.208·23-s + 0.192·27-s + 0.377·28-s − 1.29·29-s + 1.07·31-s + 0.174·33-s − 1/3·36-s + 1.31·37-s − 0.312·41-s + 0.762·43-s − 0.301·44-s + 0.875·47-s + 0.577·48-s + 1/7·49-s − 0.420·51-s + 0.412·53-s − 0.397·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
good | 2 | \( 1 + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 7 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 + 6 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 + 12 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.81766736642704411534879941192, −7.22170384564070235189575156153, −6.24539566476293518611985441220, −5.67327897139999708280650523522, −4.46643185143410760320236878209, −4.27074976927444588512621790889, −3.29069022552627357581781559287, −2.46338630580550428176993571389, −1.27798677506274336425083421134, 0,
1.27798677506274336425083421134, 2.46338630580550428176993571389, 3.29069022552627357581781559287, 4.27074976927444588512621790889, 4.46643185143410760320236878209, 5.67327897139999708280650523522, 6.24539566476293518611985441220, 7.22170384564070235189575156153, 7.81766736642704411534879941192