L(s) = 1 | − 0.982i·2-s + 3.03·4-s − 0.397·5-s + 7.22i·7-s − 6.91i·8-s + 0.390i·10-s − 12.1i·13-s + 7.10·14-s + 5.34·16-s − 20.9i·17-s + 11.1i·19-s − 1.20·20-s + 5.92·23-s − 24.8·25-s − 11.9·26-s + ⋯ |
L(s) = 1 | − 0.491i·2-s + 0.758·4-s − 0.0795·5-s + 1.03i·7-s − 0.863i·8-s + 0.0390i·10-s − 0.935i·13-s + 0.507·14-s + 0.334·16-s − 1.22i·17-s + 0.584i·19-s − 0.0603·20-s + 0.257·23-s − 0.993·25-s − 0.459·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.372 + 0.927i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.372 + 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.338330321\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.338330321\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + 0.982iT - 4T^{2} \) |
| 5 | \( 1 + 0.397T + 25T^{2} \) |
| 7 | \( 1 - 7.22iT - 49T^{2} \) |
| 13 | \( 1 + 12.1iT - 169T^{2} \) |
| 17 | \( 1 + 20.9iT - 289T^{2} \) |
| 19 | \( 1 - 11.1iT - 361T^{2} \) |
| 23 | \( 1 - 5.92T + 529T^{2} \) |
| 29 | \( 1 + 24.9iT - 841T^{2} \) |
| 31 | \( 1 - 59.6T + 961T^{2} \) |
| 37 | \( 1 + 5.96T + 1.36e3T^{2} \) |
| 41 | \( 1 + 52.0iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 17.6iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 55.3T + 2.20e3T^{2} \) |
| 53 | \( 1 - 94.6T + 2.80e3T^{2} \) |
| 59 | \( 1 - 25.1T + 3.48e3T^{2} \) |
| 61 | \( 1 - 33.4iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 94.6T + 4.48e3T^{2} \) |
| 71 | \( 1 + 82.7T + 5.04e3T^{2} \) |
| 73 | \( 1 + 46.5iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 60.3iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 150. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 134.T + 7.92e3T^{2} \) |
| 97 | \( 1 - 37.6T + 9.40e3T^{2} \) |
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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
−9.766487521228711594861500865657, −8.727378758048472585469719617033, −7.86052360432665859586282102571, −7.05558139574195565581108523341, −5.99952593056907336544788075381, −5.40769175478688254742530138206, −4.03348565161916796397183236916, −2.86135718881354388503167423672, −2.28758637373618652949034641019, −0.77486896536150257290862506551,
1.20667322360646918193004472680, 2.41137937860688600960639973991, 3.72161389489610542450966686157, 4.59817481484342642624743896892, 5.79880601721378033008168383481, 6.65243804358474132168982714078, 7.16821569672311829390566609297, 8.048717070023436791474573874844, 8.789374203176596974394062321564, 10.01995112762443036903916505811