| L(s) = 1 | + (−1.32 + 0.5i)2-s + 2.64·3-s + (1.50 − 1.32i)4-s + (−3.50 + 1.32i)6-s + 4i·7-s + (−1.32 + 2.50i)8-s + 4.00·9-s − 2.64i·11-s + (3.96 − 3.50i)12-s + (−2 − 5.29i)14-s + (0.500 − 3.96i)16-s − 3i·17-s + (−5.29 + 2.00i)18-s + 2.64i·19-s + 10.5i·21-s + (1.32 + 3.50i)22-s + ⋯ |
| L(s) = 1 | + (−0.935 + 0.353i)2-s + 1.52·3-s + (0.750 − 0.661i)4-s + (−1.42 + 0.540i)6-s + 1.51i·7-s + (−0.467 + 0.883i)8-s + 1.33·9-s − 0.797i·11-s + (1.14 − 1.01i)12-s + (−0.534 − 1.41i)14-s + (0.125 − 0.992i)16-s − 0.727i·17-s + (−1.24 + 0.471i)18-s + 0.606i·19-s + 2.30i·21-s + (0.282 + 0.746i)22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.813 - 0.581i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.813 - 0.581i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.19519 + 0.383154i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.19519 + 0.383154i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.32 - 0.5i)T \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 2.64T + 3T^{2} \) |
| 7 | \( 1 - 4iT - 7T^{2} \) |
| 11 | \( 1 + 2.64iT - 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 3iT - 17T^{2} \) |
| 19 | \( 1 - 2.64iT - 19T^{2} \) |
| 23 | \( 1 + 4iT - 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 + 10.5T + 37T^{2} \) |
| 41 | \( 1 + 5T + 41T^{2} \) |
| 43 | \( 1 - 5.29T + 43T^{2} \) |
| 47 | \( 1 + 8iT - 47T^{2} \) |
| 53 | \( 1 + 10.5T + 53T^{2} \) |
| 59 | \( 1 - 5.29iT - 59T^{2} \) |
| 61 | \( 1 + 10.5iT - 61T^{2} \) |
| 67 | \( 1 - 7.93T + 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 - 7iT - 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 + 7.93T + 83T^{2} \) |
| 89 | \( 1 - T + 89T^{2} \) |
| 97 | \( 1 + 2iT - 97T^{2} \) |
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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
−12.49464969659128353334291672713, −11.51348187918869411358837510385, −10.16760861215149777612114999159, −9.190607393385172179734590056136, −8.605537402077930067745739153173, −8.001034426001162320652754232316, −6.63224920823709166006098030981, −5.37092937476263134399023166095, −3.14325443589533872306967242419, −2.13070617041045605093388771132,
1.67095650089860251827778479311, 3.19503564901073142508403368303, 4.21054357404571717333386696193, 6.85313421774544453589408028956, 7.56656523134970228466926862886, 8.375610926513820072529665094237, 9.439018214295961896902683883830, 10.12767784143628894339140813921, 11.03243524296027662884972981176, 12.46221189743917070243469084158
