L(s) = 1 | − 3.62i·2-s − 9.10·4-s + 0.165i·5-s + 10.2·7-s + 18.4i·8-s + 0.598·10-s − 20.3·13-s − 37.2i·14-s + 30.5·16-s + 23.0i·17-s − 4.13·19-s − 1.50i·20-s + 13.5i·23-s + 24.9·25-s + 73.6i·26-s + ⋯ |
L(s) = 1 | − 1.81i·2-s − 2.27·4-s + 0.0330i·5-s + 1.47·7-s + 2.31i·8-s + 0.0598·10-s − 1.56·13-s − 2.66i·14-s + 1.90·16-s + 1.35i·17-s − 0.217·19-s − 0.0752i·20-s + 0.587i·23-s + 0.998·25-s + 2.83i·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.475886032\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.475886032\) |
\(L(2)\) |
|
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 | \( 1 + 3.62iT - 4T^{2} \) |
| 5 | \( 1 - 0.165iT - 25T^{2} \) |
| 7 | \( 1 - 10.2T + 49T^{2} \) |
| 13 | \( 1 + 20.3T + 169T^{2} \) |
| 17 | \( 1 - 23.0iT - 289T^{2} \) |
| 19 | \( 1 + 4.13T + 361T^{2} \) |
| 23 | \( 1 - 13.5iT - 529T^{2} \) |
| 29 | \( 1 - 4.01iT - 841T^{2} \) |
| 31 | \( 1 + 20.5T + 961T^{2} \) |
| 37 | \( 1 - 43.5T + 1.36e3T^{2} \) |
| 41 | \( 1 - 52.0iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 62.5T + 1.84e3T^{2} \) |
| 47 | \( 1 - 38.7iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 35.3iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 19.6iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 6.42T + 3.72e3T^{2} \) |
| 67 | \( 1 + 3.98T + 4.48e3T^{2} \) |
| 71 | \( 1 - 54.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 22.8T + 5.32e3T^{2} \) |
| 79 | \( 1 + 57.1T + 6.24e3T^{2} \) |
| 83 | \( 1 + 56.8iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 28.1iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 82.9T + 9.40e3T^{2} \) |
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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
−9.785194555966892078169170723339, −8.996197851180795057806885609632, −8.167701668198330423376134879956, −7.39707425917796620055611096628, −5.73918110613451852368902850628, −4.70185632201213811483330674339, −4.27978692301329421232085665001, −2.94080523233635655464948442551, −2.03421774846166311927508803780, −1.15089209807176981443150416925,
0.51845044468984750057909865632, 2.40823335419177358187042877728, 4.25841563520384497860632638986, 4.96521059152339138249359316347, 5.38554619422835517225585348360, 6.63439056726246779507682611436, 7.44199690991686691783020824274, 7.77866967214256013327701811590, 8.790303544675248449759046716211, 9.330102219886314608437438950158