| L(s) = 1 | − 1.18·3-s + 3.53·7-s − 1.60·9-s − 2.94·11-s − 4.01·13-s + 17-s + 6.97·19-s − 4.18·21-s − 6.12·23-s + 5.44·27-s + 5.30·29-s − 6.49·31-s + 3.48·33-s − 3.43·37-s + 4.74·39-s + 4.61·41-s + 10.2·43-s + 3.67·47-s + 5.50·49-s − 1.18·51-s + 6.77·53-s − 8.24·57-s − 9.92·59-s − 2.36·61-s − 5.66·63-s − 9.56·67-s + 7.24·69-s + ⋯ |
| L(s) = 1 | − 0.682·3-s + 1.33·7-s − 0.534·9-s − 0.888·11-s − 1.11·13-s + 0.242·17-s + 1.60·19-s − 0.912·21-s − 1.27·23-s + 1.04·27-s + 0.984·29-s − 1.16·31-s + 0.606·33-s − 0.564·37-s + 0.759·39-s + 0.720·41-s + 1.56·43-s + 0.536·47-s + 0.786·49-s − 0.165·51-s + 0.930·53-s − 1.09·57-s − 1.29·59-s − 0.302·61-s − 0.713·63-s − 1.16·67-s + 0.871·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 3 | \( 1 + 1.18T + 3T^{2} \) |
| 7 | \( 1 - 3.53T + 7T^{2} \) |
| 11 | \( 1 + 2.94T + 11T^{2} \) |
| 13 | \( 1 + 4.01T + 13T^{2} \) |
| 19 | \( 1 - 6.97T + 19T^{2} \) |
| 23 | \( 1 + 6.12T + 23T^{2} \) |
| 29 | \( 1 - 5.30T + 29T^{2} \) |
| 31 | \( 1 + 6.49T + 31T^{2} \) |
| 37 | \( 1 + 3.43T + 37T^{2} \) |
| 41 | \( 1 - 4.61T + 41T^{2} \) |
| 43 | \( 1 - 10.2T + 43T^{2} \) |
| 47 | \( 1 - 3.67T + 47T^{2} \) |
| 53 | \( 1 - 6.77T + 53T^{2} \) |
| 59 | \( 1 + 9.92T + 59T^{2} \) |
| 61 | \( 1 + 2.36T + 61T^{2} \) |
| 67 | \( 1 + 9.56T + 67T^{2} \) |
| 71 | \( 1 + 5.51T + 71T^{2} \) |
| 73 | \( 1 + 2.00T + 73T^{2} \) |
| 79 | \( 1 + 10.5T + 79T^{2} \) |
| 83 | \( 1 - 9.07T + 83T^{2} \) |
| 89 | \( 1 - 2.63T + 89T^{2} \) |
| 97 | \( 1 + 5.86T + 97T^{2} \) |
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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
−7.65153633326277655998305536946, −7.12697408990170238441240360773, −5.87381610290064325195387570447, −5.51684857575011587833097295307, −4.92222877262111973827717918288, −4.25229139881202689924247049029, −3.00983902749006884225984211707, −2.30018758362879611270537041586, −1.20154114881977391814604497764, 0,
1.20154114881977391814604497764, 2.30018758362879611270537041586, 3.00983902749006884225984211707, 4.25229139881202689924247049029, 4.92222877262111973827717918288, 5.51684857575011587833097295307, 5.87381610290064325195387570447, 7.12697408990170238441240360773, 7.65153633326277655998305536946
