| L(s) = 1 | + 0.759·2-s − 1.42·4-s + 2.86·5-s − 7-s − 2.59·8-s + 2.17·10-s + 6.96·13-s − 0.759·14-s + 0.872·16-s − 6.65·17-s + 5.75·19-s − 4.07·20-s − 0.724·23-s + 3.20·25-s + 5.28·26-s + 1.42·28-s − 7.56·29-s + 2.81·31-s + 5.86·32-s − 5.05·34-s − 2.86·35-s − 1.72·37-s + 4.36·38-s − 7.44·40-s − 1.12·41-s + 11.7·43-s − 0.550·46-s + ⋯ |
| L(s) = 1 | + 0.536·2-s − 0.711·4-s + 1.28·5-s − 0.377·7-s − 0.919·8-s + 0.688·10-s + 1.93·13-s − 0.202·14-s + 0.218·16-s − 1.61·17-s + 1.31·19-s − 0.911·20-s − 0.151·23-s + 0.641·25-s + 1.03·26-s + 0.268·28-s − 1.40·29-s + 0.506·31-s + 1.03·32-s − 0.867·34-s − 0.484·35-s − 0.283·37-s + 0.708·38-s − 1.17·40-s − 0.175·41-s + 1.78·43-s − 0.0811·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.777951824\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.777951824\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 0.759T + 2T^{2} \) |
| 5 | \( 1 - 2.86T + 5T^{2} \) |
| 13 | \( 1 - 6.96T + 13T^{2} \) |
| 17 | \( 1 + 6.65T + 17T^{2} \) |
| 19 | \( 1 - 5.75T + 19T^{2} \) |
| 23 | \( 1 + 0.724T + 23T^{2} \) |
| 29 | \( 1 + 7.56T + 29T^{2} \) |
| 31 | \( 1 - 2.81T + 31T^{2} \) |
| 37 | \( 1 + 1.72T + 37T^{2} \) |
| 41 | \( 1 + 1.12T + 41T^{2} \) |
| 43 | \( 1 - 11.7T + 43T^{2} \) |
| 47 | \( 1 - 5.18T + 47T^{2} \) |
| 53 | \( 1 + 6.25T + 53T^{2} \) |
| 59 | \( 1 - 3.98T + 59T^{2} \) |
| 61 | \( 1 - 5.97T + 61T^{2} \) |
| 67 | \( 1 - 10.3T + 67T^{2} \) |
| 71 | \( 1 + 0.995T + 71T^{2} \) |
| 73 | \( 1 + 3.53T + 73T^{2} \) |
| 79 | \( 1 - 0.669T + 79T^{2} \) |
| 83 | \( 1 + 13.4T + 83T^{2} \) |
| 89 | \( 1 - 16.6T + 89T^{2} \) |
| 97 | \( 1 - 10.1T + 97T^{2} \) |
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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
−7.974563071556000674944770175354, −6.91322638074506960002438460332, −6.21243492957374855560077434862, −5.77309956442519455652623251219, −5.24089355961935397552272827522, −4.22159774901263411263963211874, −3.68368789015929938575972025903, −2.80097336264943416579786717798, −1.83788242389732781732054587857, −0.791463000678015970976906404819,
0.791463000678015970976906404819, 1.83788242389732781732054587857, 2.80097336264943416579786717798, 3.68368789015929938575972025903, 4.22159774901263411263963211874, 5.24089355961935397552272827522, 5.77309956442519455652623251219, 6.21243492957374855560077434862, 6.91322638074506960002438460332, 7.974563071556000674944770175354
