L(s) = 1 | + 1.41·2-s + 2.00·4-s − 2.23i·5-s + 7.36i·7-s + 2.82·8-s − 3.16i·10-s − 16.6i·11-s + 9.09·13-s + 10.4i·14-s + 4.00·16-s + 4.72i·17-s − 11.5i·19-s − 4.47i·20-s − 23.5i·22-s + (18.8 − 13.2i)23-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.500·4-s − 0.447i·5-s + 1.05i·7-s + 0.353·8-s − 0.316i·10-s − 1.51i·11-s + 0.699·13-s + 0.743i·14-s + 0.250·16-s + 0.278i·17-s − 0.605i·19-s − 0.223i·20-s − 1.06i·22-s + (0.818 − 0.574i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.574 + 0.818i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.574 + 0.818i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(3.248865560\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.248865560\) |
\(L(2)\) |
|
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 | \( 1 + (-18.8 + 13.2i)T \) |
good | 7 | \( 1 - 7.36iT - 49T^{2} \) |
| 11 | \( 1 + 16.6iT - 121T^{2} \) |
| 13 | \( 1 - 9.09T + 169T^{2} \) |
| 17 | \( 1 - 4.72iT - 289T^{2} \) |
| 19 | \( 1 + 11.5iT - 361T^{2} \) |
| 29 | \( 1 - 8.88T + 841T^{2} \) |
| 31 | \( 1 + 45.3T + 961T^{2} \) |
| 37 | \( 1 - 9.59iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 20.7T + 1.68e3T^{2} \) |
| 43 | \( 1 + 21.4iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 31.0T + 2.20e3T^{2} \) |
| 53 | \( 1 + 70.7iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 47.6T + 3.48e3T^{2} \) |
| 61 | \( 1 + 16.1iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 11.8iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 2.91T + 5.04e3T^{2} \) |
| 73 | \( 1 - 136.T + 5.32e3T^{2} \) |
| 79 | \( 1 + 37.4iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 108. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 10.9iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 70.0iT - 9.40e3T^{2} \) |
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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
−8.648578925349673876532505527359, −8.329634910585617471351453957079, −7.06413701706854919889553972234, −6.20614487878016950610110014438, −5.58511432575780849425334105863, −4.97015105432912427907050564352, −3.78315047603347557570372431781, −3.06389689531456391962256976390, −2.01057528314073615730527900114, −0.67487368786853785138998249325,
1.19822501919515418373833011414, 2.25410126690932585446861020803, 3.47359211343747309492374963185, 4.07533405776679544988410222771, 4.94102746287653190370455930881, 5.84148018825431047815376419361, 6.91042425836038586584965521698, 7.20521924673870335909570841048, 7.983681536963234059539421604578, 9.213628009611328131729732263041