L(s) = 1 | − 2.23i·3-s − 2.00·9-s − 2.23i·11-s + 4i·13-s + 3·17-s + 2.23i·19-s − 8.94·23-s − 2.23i·27-s + 4i·29-s − 8.94·31-s − 5.00·33-s − 8i·37-s + 8.94·39-s − 5·41-s − 8.94i·43-s + ⋯ |
L(s) = 1 | − 1.29i·3-s − 0.666·9-s − 0.674i·11-s + 1.10i·13-s + 0.727·17-s + 0.512i·19-s − 1.86·23-s − 0.430i·27-s + 0.742i·29-s − 1.60·31-s − 0.870·33-s − 1.31i·37-s + 1.43·39-s − 0.780·41-s − 1.36i·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3350024417\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3350024417\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + 2.23iT - 3T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + 2.23iT - 11T^{2} \) |
| 13 | \( 1 - 4iT - 13T^{2} \) |
| 17 | \( 1 - 3T + 17T^{2} \) |
| 19 | \( 1 - 2.23iT - 19T^{2} \) |
| 23 | \( 1 + 8.94T + 23T^{2} \) |
| 29 | \( 1 - 4iT - 29T^{2} \) |
| 31 | \( 1 + 8.94T + 31T^{2} \) |
| 37 | \( 1 + 8iT - 37T^{2} \) |
| 41 | \( 1 + 5T + 41T^{2} \) |
| 43 | \( 1 + 8.94iT - 43T^{2} \) |
| 47 | \( 1 + 8.94T + 47T^{2} \) |
| 53 | \( 1 + 4iT - 53T^{2} \) |
| 59 | \( 1 - 8.94iT - 59T^{2} \) |
| 61 | \( 1 - 8iT - 61T^{2} \) |
| 67 | \( 1 + 6.70iT - 67T^{2} \) |
| 71 | \( 1 - 8.94T + 71T^{2} \) |
| 73 | \( 1 + 9T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 6.70iT - 83T^{2} \) |
| 89 | \( 1 + 15T + 89T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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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
−8.105516709877738936966039688493, −7.38858548123732639457608124728, −6.81766383035555118767465060564, −6.00833891248269748870059787458, −5.44751040203606988422813418196, −4.13631515732100051630945210583, −3.38881241923579262773212645473, −2.05374495876676979965081198916, −1.54639760062492613760155207057, −0.096441633202889356990664552413,
1.68637649933675878216714777828, 2.98933998591485393410476198986, 3.68355747511223221537375011591, 4.52718034084306128463736457329, 5.17472806567813325068534705768, 5.90869067126441523385798884196, 6.86348330333939407034843505973, 7.929250506954278807879208690027, 8.262555055294317868795733624151, 9.563434041274885829118852970273