| L(s) = 1 | − 1.73·3-s − 11-s − 1.73·13-s + 5.19·17-s − 2.82·19-s + 2.44·23-s + 5.19·27-s − 7·29-s + 7.07·31-s + 1.73·33-s − 7.34·37-s + 2.99·39-s + 7.07·41-s − 9.79·43-s + 12.1·47-s − 9·51-s + 12.2·53-s + 4.89·57-s − 7.07·59-s − 14.1·61-s + 12.2·67-s − 4.24·69-s − 10·71-s + 3·79-s − 9·81-s + 12.1·87-s − 12.2·93-s + ⋯ |
| L(s) = 1 | − 1.00·3-s − 0.301·11-s − 0.480·13-s + 1.26·17-s − 0.648·19-s + 0.510·23-s + 1.00·27-s − 1.29·29-s + 1.27·31-s + 0.301·33-s − 1.20·37-s + 0.480·39-s + 1.10·41-s − 1.49·43-s + 1.76·47-s − 1.26·51-s + 1.68·53-s + 0.648·57-s − 0.920·59-s − 1.81·61-s + 1.49·67-s − 0.510·69-s − 1.18·71-s + 0.337·79-s − 81-s + 1.29·87-s − 1.27·93-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 1.73T + 3T^{2} \) |
| 11 | \( 1 + T + 11T^{2} \) |
| 13 | \( 1 + 1.73T + 13T^{2} \) |
| 17 | \( 1 - 5.19T + 17T^{2} \) |
| 19 | \( 1 + 2.82T + 19T^{2} \) |
| 23 | \( 1 - 2.44T + 23T^{2} \) |
| 29 | \( 1 + 7T + 29T^{2} \) |
| 31 | \( 1 - 7.07T + 31T^{2} \) |
| 37 | \( 1 + 7.34T + 37T^{2} \) |
| 41 | \( 1 - 7.07T + 41T^{2} \) |
| 43 | \( 1 + 9.79T + 43T^{2} \) |
| 47 | \( 1 - 12.1T + 47T^{2} \) |
| 53 | \( 1 - 12.2T + 53T^{2} \) |
| 59 | \( 1 + 7.07T + 59T^{2} \) |
| 61 | \( 1 + 14.1T + 61T^{2} \) |
| 67 | \( 1 - 12.2T + 67T^{2} \) |
| 71 | \( 1 + 10T + 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 - 3T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 5.19T + 97T^{2} \) |
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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
−7.78378290659533680234247257438, −7.14545943690616802187359127353, −6.34206104417966657362976645800, −5.63010319988242557387410955517, −5.16990087834822778085302529534, −4.32225828240741342126017424633, −3.31526204908497531362347969748, −2.40628849041128665087918945378, −1.13888460701417521227659848887, 0,
1.13888460701417521227659848887, 2.40628849041128665087918945378, 3.31526204908497531362347969748, 4.32225828240741342126017424633, 5.16990087834822778085302529534, 5.63010319988242557387410955517, 6.34206104417966657362976645800, 7.14545943690616802187359127353, 7.78378290659533680234247257438
