| L(s) = 1 | + (1.29 − 0.577i)2-s + (2.25 + 0.885i)3-s + (1.33 − 1.49i)4-s + (0.346 − 0.0521i)5-s + (3.42 − 0.161i)6-s + (−2.27 + 1.35i)7-s + (0.856 − 2.69i)8-s + (2.10 + 1.95i)9-s + (0.416 − 0.267i)10-s + (−1.49 + 1.38i)11-s + (4.32 − 2.18i)12-s + (−0.125 − 0.550i)13-s + (−2.15 + 3.06i)14-s + (0.827 + 0.188i)15-s + (−0.452 − 3.97i)16-s + (3.72 − 0.279i)17-s + ⋯ |
| L(s) = 1 | + (0.912 − 0.408i)2-s + (1.30 + 0.511i)3-s + (0.665 − 0.746i)4-s + (0.154 − 0.0233i)5-s + (1.39 − 0.0658i)6-s + (−0.859 + 0.511i)7-s + (0.302 − 0.953i)8-s + (0.703 + 0.652i)9-s + (0.131 − 0.0845i)10-s + (−0.450 + 0.418i)11-s + (1.24 − 0.631i)12-s + (−0.0348 − 0.152i)13-s + (−0.574 + 0.818i)14-s + (0.213 + 0.0487i)15-s + (−0.113 − 0.993i)16-s + (0.903 − 0.0676i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.971 + 0.235i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.971 + 0.235i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.08006 - 0.367285i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.08006 - 0.367285i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.29 + 0.577i)T \) |
| 7 | \( 1 + (2.27 - 1.35i)T \) |
| good | 3 | \( 1 + (-2.25 - 0.885i)T + (2.19 + 2.04i)T^{2} \) |
| 5 | \( 1 + (-0.346 + 0.0521i)T + (4.77 - 1.47i)T^{2} \) |
| 11 | \( 1 + (1.49 - 1.38i)T + (0.822 - 10.9i)T^{2} \) |
| 13 | \( 1 + (0.125 + 0.550i)T + (-11.7 + 5.64i)T^{2} \) |
| 17 | \( 1 + (-3.72 + 0.279i)T + (16.8 - 2.53i)T^{2} \) |
| 19 | \( 1 + (3.51 + 2.02i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.49 + 0.111i)T + (22.7 + 3.42i)T^{2} \) |
| 29 | \( 1 + (1.23 - 2.55i)T + (-18.0 - 22.6i)T^{2} \) |
| 31 | \( 1 + (-0.563 - 0.975i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.18 - 4.66i)T + (-13.5 + 34.4i)T^{2} \) |
| 41 | \( 1 + (-3.61 + 2.88i)T + (9.12 - 39.9i)T^{2} \) |
| 43 | \( 1 + (0.185 - 0.232i)T + (-9.56 - 41.9i)T^{2} \) |
| 47 | \( 1 + (6.30 + 1.94i)T + (38.8 + 26.4i)T^{2} \) |
| 53 | \( 1 + (2.64 - 3.87i)T + (-19.3 - 49.3i)T^{2} \) |
| 59 | \( 1 + (0.580 - 3.85i)T + (-56.3 - 17.3i)T^{2} \) |
| 61 | \( 1 + (-12.1 + 8.29i)T + (22.2 - 56.7i)T^{2} \) |
| 67 | \( 1 + (-4.11 - 7.11i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (0.343 + 0.713i)T + (-44.2 + 55.5i)T^{2} \) |
| 73 | \( 1 + (-0.852 - 2.76i)T + (-60.3 + 41.1i)T^{2} \) |
| 79 | \( 1 + (-1.15 - 0.669i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-5.93 - 1.35i)T + (74.7 + 36.0i)T^{2} \) |
| 89 | \( 1 + (10.5 - 11.3i)T + (-6.65 - 88.7i)T^{2} \) |
| 97 | \( 1 + 15.4iT - 97T^{2} \) |
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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
−11.34538157044065998754960391283, −10.03397241381369146523821535012, −9.779270883450520637992714639678, −8.687447903001934962501820676654, −7.55201982283103322044430999562, −6.32823316919853729810662305574, −5.24517552874776399731719360219, −3.98996651331611118407846315684, −3.08950804888932825707043191859, −2.21064823994449268732388564797,
2.17496373472481511701003134382, 3.24613596604243305098562006515, 4.07268223298027981938709698895, 5.72079021818852700951382446338, 6.60808725193585312514194705871, 7.71771948340652596400989646045, 8.136288133644336784432937022335, 9.381799191978516213178720840489, 10.37626030079112572029997176126, 11.60799170082855747862769783761
