L(s) = 1 | − 2.15·2-s + 2.64·4-s + 5-s − 2.12·7-s − 1.38·8-s − 2.15·10-s − 0.985·13-s + 4.57·14-s − 2.30·16-s + 7.71·17-s + 2·19-s + 2.64·20-s + 7.82·23-s + 25-s + 2.12·26-s − 5.61·28-s − 1.85·29-s + 6.24·31-s + 7.72·32-s − 16.6·34-s − 2.12·35-s − 5.71·37-s − 4.30·38-s − 1.38·40-s − 3.47·41-s − 3.96·43-s − 16.8·46-s + ⋯ |
L(s) = 1 | − 1.52·2-s + 1.32·4-s + 0.447·5-s − 0.802·7-s − 0.489·8-s − 0.681·10-s − 0.273·13-s + 1.22·14-s − 0.575·16-s + 1.87·17-s + 0.458·19-s + 0.590·20-s + 1.63·23-s + 0.200·25-s + 0.416·26-s − 1.06·28-s − 0.345·29-s + 1.12·31-s + 1.36·32-s − 2.85·34-s − 0.359·35-s − 0.939·37-s − 0.699·38-s − 0.218·40-s − 0.543·41-s − 0.605·43-s − 2.48·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8841501379\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8841501379\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 89 | \( 1 - T \) |
good | 2 | \( 1 + 2.15T + 2T^{2} \) |
| 7 | \( 1 + 2.12T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 0.985T + 13T^{2} \) |
| 17 | \( 1 - 7.71T + 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 - 7.82T + 23T^{2} \) |
| 29 | \( 1 + 1.85T + 29T^{2} \) |
| 31 | \( 1 - 6.24T + 31T^{2} \) |
| 37 | \( 1 + 5.71T + 37T^{2} \) |
| 41 | \( 1 + 3.47T + 41T^{2} \) |
| 43 | \( 1 + 3.96T + 43T^{2} \) |
| 47 | \( 1 + 3.47T + 47T^{2} \) |
| 53 | \( 1 - 2.33T + 53T^{2} \) |
| 59 | \( 1 - 4.84T + 59T^{2} \) |
| 61 | \( 1 + 0.503T + 61T^{2} \) |
| 67 | \( 1 + 1.94T + 67T^{2} \) |
| 71 | \( 1 + 1.78T + 71T^{2} \) |
| 73 | \( 1 - 15.8T + 73T^{2} \) |
| 79 | \( 1 + 7.92T + 79T^{2} \) |
| 83 | \( 1 - 12.0T + 83T^{2} \) |
| 97 | \( 1 - 0.819T + 97T^{2} \) |
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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
−8.470056339519521276414664632087, −7.913629430212444382923290653410, −7.04499888723949883870041526952, −6.65647677247724816896168379988, −5.60019039315750713629927834204, −4.89640712423926009023266878606, −3.45401377505032646725156415551, −2.79667221025084835965051893401, −1.57924656397591606048666994316, −0.71698288404846958156307271099,
0.71698288404846958156307271099, 1.57924656397591606048666994316, 2.79667221025084835965051893401, 3.45401377505032646725156415551, 4.89640712423926009023266878606, 5.60019039315750713629927834204, 6.65647677247724816896168379988, 7.04499888723949883870041526952, 7.913629430212444382923290653410, 8.470056339519521276414664632087