| L(s) = 1 | − 2-s + 4-s + 3.80·5-s − 8-s − 3.80·10-s − 11-s + 6.44·13-s + 16-s + 3.80·17-s − 6.64·19-s + 3.80·20-s + 22-s − 0.842·23-s + 9.44·25-s − 6.44·26-s + 8.44·29-s + 9.40·31-s − 32-s − 3.80·34-s + 9.60·37-s + 6.64·38-s − 3.80·40-s − 3.80·41-s − 8.04·43-s − 44-s + 0.842·46-s + 6.24·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 1.69·5-s − 0.353·8-s − 1.20·10-s − 0.301·11-s + 1.78·13-s + 0.250·16-s + 0.921·17-s − 1.52·19-s + 0.849·20-s + 0.213·22-s − 0.175·23-s + 1.88·25-s − 1.26·26-s + 1.56·29-s + 1.68·31-s − 0.176·32-s − 0.651·34-s + 1.57·37-s + 1.07·38-s − 0.600·40-s − 0.593·41-s − 1.22·43-s − 0.150·44-s + 0.124·46-s + 0.910·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.600514695\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.600514695\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 5 | \( 1 - 3.80T + 5T^{2} \) |
| 13 | \( 1 - 6.44T + 13T^{2} \) |
| 17 | \( 1 - 3.80T + 17T^{2} \) |
| 19 | \( 1 + 6.64T + 19T^{2} \) |
| 23 | \( 1 + 0.842T + 23T^{2} \) |
| 29 | \( 1 - 8.44T + 29T^{2} \) |
| 31 | \( 1 - 9.40T + 31T^{2} \) |
| 37 | \( 1 - 9.60T + 37T^{2} \) |
| 41 | \( 1 + 3.80T + 41T^{2} \) |
| 43 | \( 1 + 8.04T + 43T^{2} \) |
| 47 | \( 1 - 6.24T + 47T^{2} \) |
| 53 | \( 1 - 1.60T + 53T^{2} \) |
| 59 | \( 1 + 9.28T + 59T^{2} \) |
| 61 | \( 1 + 8.75T + 61T^{2} \) |
| 67 | \( 1 - 1.15T + 67T^{2} \) |
| 71 | \( 1 - 6.75T + 71T^{2} \) |
| 73 | \( 1 - 7.80T + 73T^{2} \) |
| 79 | \( 1 - 2.84T + 79T^{2} \) |
| 83 | \( 1 + 9.80T + 83T^{2} \) |
| 89 | \( 1 + 5.15T + 89T^{2} \) |
| 97 | \( 1 + 18.8T + 97T^{2} \) |
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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
−7.967802626161871849237274952151, −6.75295771742430111756963094489, −6.19212683101293836969818697778, −6.06694228325894211912110081807, −5.09252852227894712585889647080, −4.22887882851290003534824125179, −3.09256463906592157780613195373, −2.46594073414022019442698184308, −1.55700416565525383807665476334, −0.937319188773677678993801308121,
0.937319188773677678993801308121, 1.55700416565525383807665476334, 2.46594073414022019442698184308, 3.09256463906592157780613195373, 4.22887882851290003534824125179, 5.09252852227894712585889647080, 6.06694228325894211912110081807, 6.19212683101293836969818697778, 6.75295771742430111756963094489, 7.967802626161871849237274952151
