| L(s) = 1 | + i·2-s − 4-s + 0.267i·5-s + 0.732i·7-s − i·8-s − 0.267·10-s − 4.73i·11-s − 0.732·14-s + 16-s − 2.26·17-s − 1.26i·19-s − 0.267i·20-s + 4.73·22-s − 6.19·23-s + 4.92·25-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.5·4-s + 0.119i·5-s + 0.276i·7-s − 0.353i·8-s − 0.0847·10-s − 1.42i·11-s − 0.195·14-s + 0.250·16-s − 0.550·17-s − 0.290i·19-s − 0.0599i·20-s + 1.00·22-s − 1.29·23-s + 0.985·25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.960 - 0.277i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.960 - 0.277i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8026172892\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8026172892\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 - 0.267iT - 5T^{2} \) |
| 7 | \( 1 - 0.732iT - 7T^{2} \) |
| 11 | \( 1 + 4.73iT - 11T^{2} \) |
| 17 | \( 1 + 2.26T + 17T^{2} \) |
| 19 | \( 1 + 1.26iT - 19T^{2} \) |
| 23 | \( 1 + 6.19T + 23T^{2} \) |
| 29 | \( 1 + 2.46T + 29T^{2} \) |
| 31 | \( 1 - 5.46iT - 31T^{2} \) |
| 37 | \( 1 - 10.4iT - 37T^{2} \) |
| 41 | \( 1 - 11.3iT - 41T^{2} \) |
| 43 | \( 1 + 7.66T + 43T^{2} \) |
| 47 | \( 1 - 8.19iT - 47T^{2} \) |
| 53 | \( 1 + 0.464T + 53T^{2} \) |
| 59 | \( 1 + 8iT - 59T^{2} \) |
| 61 | \( 1 - 1.19T + 61T^{2} \) |
| 67 | \( 1 - 11.1iT - 67T^{2} \) |
| 71 | \( 1 - 1.26iT - 71T^{2} \) |
| 73 | \( 1 + 9.73iT - 73T^{2} \) |
| 79 | \( 1 + 9.46T + 79T^{2} \) |
| 83 | \( 1 - 10.1iT - 83T^{2} \) |
| 89 | \( 1 - 2.53iT - 89T^{2} \) |
| 97 | \( 1 - 6iT - 97T^{2} \) |
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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
−8.716727167578236702309870022480, −8.418488972189044625916330325170, −7.62209353850871041828246016213, −6.54685230359560823149645254826, −6.26083451395660273360724818217, −5.30487877025916276875374820309, −4.58682070005904547483214069612, −3.51452518265488197486432134784, −2.73994971371760346649848996072, −1.22908884002359873655489298257,
0.26085182800569717763175867203, 1.79636515511229659383477890065, 2.36840272403895561543210616928, 3.75127391137503807564301600127, 4.24551169775977822562442416109, 5.13249255116510348281264422059, 5.97920201980613615158359463612, 7.09094581939577671757210439272, 7.53862578667208030779271220572, 8.594481150303802868182784752399
