L(s) = 1 | + i·2-s − 4-s + 3.06i·5-s + (2.37 − 1.16i)7-s − i·8-s − 3.06·10-s + (3.20 + 0.857i)11-s + 0.338·13-s + (1.16 + 2.37i)14-s + 16-s + 0.314·17-s + 6.09·19-s − 3.06i·20-s + (−0.857 + 3.20i)22-s + 3.37·23-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 0.5·4-s + 1.36i·5-s + (0.897 − 0.441i)7-s − 0.353i·8-s − 0.968·10-s + (0.966 + 0.258i)11-s + 0.0939·13-s + (0.312 + 0.634i)14-s + 0.250·16-s + 0.0762·17-s + 1.39·19-s − 0.684i·20-s + (−0.182 + 0.683i)22-s + 0.703·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.195 - 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.195 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.934296645\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.934296645\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-2.37 + 1.16i)T \) |
| 11 | \( 1 + (-3.20 - 0.857i)T \) |
good | 5 | \( 1 - 3.06iT - 5T^{2} \) |
| 13 | \( 1 - 0.338T + 13T^{2} \) |
| 17 | \( 1 - 0.314T + 17T^{2} \) |
| 19 | \( 1 - 6.09T + 19T^{2} \) |
| 23 | \( 1 - 3.37T + 23T^{2} \) |
| 29 | \( 1 + 4.40iT - 29T^{2} \) |
| 31 | \( 1 + 0.722iT - 31T^{2} \) |
| 37 | \( 1 + 7.49T + 37T^{2} \) |
| 41 | \( 1 - 8.09T + 41T^{2} \) |
| 43 | \( 1 - 4.12iT - 43T^{2} \) |
| 47 | \( 1 - 8.77iT - 47T^{2} \) |
| 53 | \( 1 + 13.5T + 53T^{2} \) |
| 59 | \( 1 - 4.67iT - 59T^{2} \) |
| 61 | \( 1 - 13.7T + 61T^{2} \) |
| 67 | \( 1 + 1.73T + 67T^{2} \) |
| 71 | \( 1 + 13.5T + 71T^{2} \) |
| 73 | \( 1 + 1.68T + 73T^{2} \) |
| 79 | \( 1 + 6.86iT - 79T^{2} \) |
| 83 | \( 1 + 7.11T + 83T^{2} \) |
| 89 | \( 1 + 2.28iT - 89T^{2} \) |
| 97 | \( 1 - 5.08iT - 97T^{2} \) |
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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
−9.737710254567874838512287668313, −8.984239713899964996255940944654, −7.84546936869375304995963297598, −7.35999119384078522696139908020, −6.67524673683373235367194909541, −5.85260174774581762894976965786, −4.78827406232670172505739667369, −3.86609287095921048011132894518, −2.87578152711925302398085513028, −1.33240927364703271562904557979,
0.991253240575406569018903305665, 1.71735484921011989379228849115, 3.21180686640055451465132441381, 4.25130568702846784281418802628, 5.11677063325223633154307444009, 5.57665131075024776679866867305, 7.01212871892478192100508223710, 8.062800300718378117833225954892, 8.844988223163179686126066756050, 9.115270904133268249462074516469