| L(s) = 1 | + 1.78·2-s + 1.17·4-s − 0.267·5-s + 2.76·7-s − 1.47·8-s − 0.476·10-s − 3.66·11-s − 0.303·13-s + 4.91·14-s − 4.97·16-s + 2.93·17-s − 3.32·19-s − 0.313·20-s − 6.52·22-s + 23-s − 4.92·25-s − 0.540·26-s + 3.23·28-s − 29-s − 2.17·31-s − 5.89·32-s + 5.23·34-s − 0.739·35-s + 3.32·37-s − 5.92·38-s + 0.395·40-s + 5.47·41-s + ⋯ |
| L(s) = 1 | + 1.25·2-s + 0.585·4-s − 0.119·5-s + 1.04·7-s − 0.522·8-s − 0.150·10-s − 1.10·11-s − 0.0842·13-s + 1.31·14-s − 1.24·16-s + 0.712·17-s − 0.763·19-s − 0.0700·20-s − 1.39·22-s + 0.208·23-s − 0.985·25-s − 0.106·26-s + 0.611·28-s − 0.185·29-s − 0.389·31-s − 1.04·32-s + 0.897·34-s − 0.125·35-s + 0.547·37-s − 0.961·38-s + 0.0625·40-s + 0.854·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 - 1.78T + 2T^{2} \) |
| 5 | \( 1 + 0.267T + 5T^{2} \) |
| 7 | \( 1 - 2.76T + 7T^{2} \) |
| 11 | \( 1 + 3.66T + 11T^{2} \) |
| 13 | \( 1 + 0.303T + 13T^{2} \) |
| 17 | \( 1 - 2.93T + 17T^{2} \) |
| 19 | \( 1 + 3.32T + 19T^{2} \) |
| 31 | \( 1 + 2.17T + 31T^{2} \) |
| 37 | \( 1 - 3.32T + 37T^{2} \) |
| 41 | \( 1 - 5.47T + 41T^{2} \) |
| 43 | \( 1 + 1.76T + 43T^{2} \) |
| 47 | \( 1 + 2.08T + 47T^{2} \) |
| 53 | \( 1 + 1.14T + 53T^{2} \) |
| 59 | \( 1 + 9.61T + 59T^{2} \) |
| 61 | \( 1 + 9.75T + 61T^{2} \) |
| 67 | \( 1 + 9.96T + 67T^{2} \) |
| 71 | \( 1 + 3.49T + 71T^{2} \) |
| 73 | \( 1 + 6.63T + 73T^{2} \) |
| 79 | \( 1 - 15.8T + 79T^{2} \) |
| 83 | \( 1 + 3.96T + 83T^{2} \) |
| 89 | \( 1 + 15.6T + 89T^{2} \) |
| 97 | \( 1 + 1.64T + 97T^{2} \) |
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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
−7.82462401793039880430753916286, −6.91684168250153529375287359108, −5.88284376365864435226729168134, −5.56618998433076420591933549583, −4.68096965349712147285377548538, −4.32748838407170048739601344993, −3.32497645511216660762199344174, −2.58220056777314962657265228697, −1.63976905987317479622226868631, 0,
1.63976905987317479622226868631, 2.58220056777314962657265228697, 3.32497645511216660762199344174, 4.32748838407170048739601344993, 4.68096965349712147285377548538, 5.56618998433076420591933549583, 5.88284376365864435226729168134, 6.91684168250153529375287359108, 7.82462401793039880430753916286
