L(s) = 1 | + i·3-s − 7-s − 9-s + i·11-s − i·13-s − 17-s − i·19-s − i·21-s − 23-s − i·27-s + i·29-s − 31-s − 33-s + i·37-s + 39-s + ⋯ |
L(s) = 1 | + i·3-s − 7-s − 9-s + i·11-s − i·13-s − 17-s − i·19-s − i·21-s − 23-s − i·27-s + i·29-s − 31-s − 33-s + i·37-s + 39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.06050106139 + 0.3041593752i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.06050106139 + 0.3041593752i\) |
\(L(1)\) |
\(\approx\) |
\(0.6490082549 + 0.2688280213i\) |
\(L(1)\) |
\(\approx\) |
\(0.6490082549 + 0.2688280213i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + T \) |
| 7 | \( 1 + iT \) |
| 11 | \( 1 \) |
| 13 | \( 1 \) |
| 17 | \( 1 \) |
| 19 | \( 1 - T \) |
| 23 | \( 1 \) |
| 29 | \( 1 - T \) |
| 31 | \( 1 \) |
| 37 | \( 1 + iT \) |
| 41 | \( 1 \) |
| 43 | \( 1 - iT \) |
| 47 | \( 1 \) |
| 53 | \( 1 \) |
| 59 | \( 1 \) |
| 61 | \( 1 - T \) |
| 67 | \( 1 \) |
| 71 | \( 1 - iT \) |
| 73 | \( 1 \) |
| 79 | \( 1 - iT \) |
| 83 | \( 1 \) |
| 89 | \( 1 - T \) |
| 97 | \( 1 \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−30.054765179056206859261430109545, −29.154159444805894128087749276469, −28.51795806731422008873390517175, −26.77439568377164270801753785586, −25.870047229084608614769333256824, −24.73045967566660650225496406032, −23.858054673733082498876048453350, −22.77234370120555460659974227435, −21.68343617014007170731203535082, −20.08266926545199931031108919019, −19.147867432423578633105717059133, −18.38463001490553815179797928561, −16.94377096047829303010143215725, −15.995512051187264359553029930803, −14.23228950840737157581629899140, −13.370924599599317017520944708372, −12.26290315233048464977216954868, −11.112812824637837644876221934105, −9.41590537706542804344715977116, −8.18456517199584090192804828365, −6.74028866790318099628048572604, −5.88981876392024743960749689627, −3.68087883657849355729892612389, −2.07675048939256145975620763136, −0.13853650194408508948009201069,
2.71794826130170972459527086965, 4.09769813699763886205507389319, 5.45881613904210370021931340275, 6.964059069131927889310429754134, 8.74821524100515707299788341949, 9.82206201037914464819291395220, 10.74836471424836936082390488716, 12.29319032108009203323924958783, 13.53685091930915154448259132396, 15.13059340113388784236150144731, 15.72356553163442059323948050815, 16.9777548637102204668459949360, 18.093788426827801775443253356595, 19.91969291626378757328000731117, 20.26258910072587233897931977120, 22.008620328360075262707684951018, 22.4079917877188291524873447196, 23.68598838114093829757165996084, 25.411595155682671673588879220606, 25.97301765456541176812088132358, 27.184866223128984334025766963765, 28.1888168620164840246988983474, 29.01169317334837132170979913394, 30.43401159894429032474277119489, 31.612913252279338373464364516012