L(s) = 1 | − 1.41·2-s + 2.00·4-s + 2.23i·5-s + 7.10i·7-s − 2.82·8-s − 3.16i·10-s − 11.2i·11-s + 20.0·13-s − 10.0i·14-s + 4.00·16-s − 1.63i·17-s + 29.4i·19-s + 4.47i·20-s + 15.9i·22-s + (−20.0 − 11.3i)23-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.500·4-s + 0.447i·5-s + 1.01i·7-s − 0.353·8-s − 0.316i·10-s − 1.02i·11-s + 1.54·13-s − 0.717i·14-s + 0.250·16-s − 0.0959i·17-s + 1.54i·19-s + 0.223i·20-s + 0.724i·22-s + (−0.870 − 0.491i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.491 - 0.870i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.491 - 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.543251655\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.543251655\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 1.41T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - 2.23iT \) |
| 23 | \( 1 + (20.0 + 11.3i)T \) |
good | 7 | \( 1 - 7.10iT - 49T^{2} \) |
| 11 | \( 1 + 11.2iT - 121T^{2} \) |
| 13 | \( 1 - 20.0T + 169T^{2} \) |
| 17 | \( 1 + 1.63iT - 289T^{2} \) |
| 19 | \( 1 - 29.4iT - 361T^{2} \) |
| 29 | \( 1 - 50.3T + 841T^{2} \) |
| 31 | \( 1 - 11.1T + 961T^{2} \) |
| 37 | \( 1 - 40.5iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 7.24T + 1.68e3T^{2} \) |
| 43 | \( 1 + 71.7iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 6.40T + 2.20e3T^{2} \) |
| 53 | \( 1 - 20.4iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 65.8T + 3.48e3T^{2} \) |
| 61 | \( 1 + 37.7iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 124. iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 43.5T + 5.04e3T^{2} \) |
| 73 | \( 1 - 48.1T + 5.32e3T^{2} \) |
| 79 | \( 1 - 101. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 102. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 9.63iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 143. iT - 9.40e3T^{2} \) |
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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
−8.764534681720985903665019522447, −8.462539875901069473666986494611, −7.84774010586838870491697504864, −6.43591484427188270586596744934, −6.21589045476311167069866664639, −5.37978359641737342781227992654, −3.88915075987919366201954465695, −3.11324914779772748074257417683, −2.09604418328398600738167707045, −0.930548317173473956013968382010,
0.65111245857275797809021864325, 1.46727300270508758068319609885, 2.72752436647597891457898565228, 3.98370503742171674298022158554, 4.58618060583428413051437961131, 5.78953911274273433479986729202, 6.70162049680144554076689706394, 7.26479811749690336736489129537, 8.168191282800868504833098065415, 8.736727919714423949608290264756