| L(s) = 1 | + (−0.342 + 0.939i)5-s + (−0.342 − 0.939i)11-s + (0.984 − 0.173i)13-s + 17-s − i·19-s + (0.173 + 0.984i)23-s + (−0.766 − 0.642i)25-s + (−0.984 − 0.173i)29-s + (−0.766 + 0.642i)31-s + (0.866 + 0.5i)37-s + (−0.173 − 0.984i)41-s + (0.642 − 0.766i)43-s + (0.766 + 0.642i)47-s + (−0.866 − 0.5i)53-s + 55-s + ⋯ |
| L(s) = 1 | + (−0.342 + 0.939i)5-s + (−0.342 − 0.939i)11-s + (0.984 − 0.173i)13-s + 17-s − i·19-s + (0.173 + 0.984i)23-s + (−0.766 − 0.642i)25-s + (−0.984 − 0.173i)29-s + (−0.766 + 0.642i)31-s + (0.866 + 0.5i)37-s + (−0.173 − 0.984i)41-s + (0.642 − 0.766i)43-s + (0.766 + 0.642i)47-s + (−0.866 − 0.5i)53-s + 55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.749 - 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.749 - 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.322982017 - 0.5002949722i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.322982017 - 0.5002949722i\) |
| \(L(1)\) |
\(\approx\) |
\(1.022286767 + 0.005286230390i\) |
| \(L(1)\) |
\(\approx\) |
\(1.022286767 + 0.005286230390i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + (-0.342 + 0.939i)T \) |
| 11 | \( 1 + (-0.342 - 0.939i)T \) |
| 13 | \( 1 + (0.984 - 0.173i)T \) |
| 17 | \( 1 + T \) |
| 19 | \( 1 - iT \) |
| 23 | \( 1 + (0.173 + 0.984i)T \) |
| 29 | \( 1 + (-0.984 - 0.173i)T \) |
| 31 | \( 1 + (-0.766 + 0.642i)T \) |
| 37 | \( 1 + (0.866 + 0.5i)T \) |
| 41 | \( 1 + (-0.173 - 0.984i)T \) |
| 43 | \( 1 + (0.642 - 0.766i)T \) |
| 47 | \( 1 + (0.766 + 0.642i)T \) |
| 53 | \( 1 + (-0.866 - 0.5i)T \) |
| 59 | \( 1 + (0.984 - 0.173i)T \) |
| 61 | \( 1 + (0.642 - 0.766i)T \) |
| 67 | \( 1 + (0.342 - 0.939i)T \) |
| 71 | \( 1 + (-0.5 - 0.866i)T \) |
| 73 | \( 1 + (-0.5 - 0.866i)T \) |
| 79 | \( 1 + (-0.939 + 0.342i)T \) |
| 83 | \( 1 + (0.984 + 0.173i)T \) |
| 89 | \( 1 - T \) |
| 97 | \( 1 + (-0.766 - 0.642i)T \) |
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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
−18.95831427756626057035587774841, −18.56243333719560804244731085015, −17.74862181536732489506188180315, −16.83765065646181063472389103374, −16.37856840237351777826468834864, −15.8338309880406532722487817803, −14.7945620357063019044433123562, −14.459180240692834308046249348465, −13.1606558632025577171834820864, −12.91467120482638765482185885275, −12.1403584961048023810876023816, −11.456041462268616613823075923503, −10.59797889637634305949567357300, −9.76963063847505552158686759832, −9.16656800555792794049172142496, −8.25714478324475034512067374823, −7.78857466003725252628496953968, −6.94989414498649695115363700417, −5.80366769886067460227170436125, −5.4089046719772445095033569477, −4.22335432663872478633145840483, −3.9665805174371189680568195579, −2.74803733987498909103513078002, −1.688711767898914066869889390400, −0.9812671667323281573808208064,
0.50817300173972928229910508819, 1.65666519664286018472238118874, 2.80052490428290892341510857754, 3.3943134751443203898386445802, 3.98947451429137330753602636182, 5.32222808008002827283679263461, 5.82515092794988999265597910465, 6.70389339792160859532013075184, 7.47562950615670573932049002931, 8.075102269427640495393418794, 8.948892399913998802533866223000, 9.71755390263094100098985219870, 10.74711779709428096391717065891, 11.031424507603961256223107642774, 11.70077156528064486303953158061, 12.67945415336363700483921209244, 13.50875180211010258706081532374, 13.994283817847754472751531196047, 14.80323080629557859175366967453, 15.557406474420258142612953409303, 16.002664914463593982760912892930, 16.85821948594727476067992292715, 17.726506539017292265802202126126, 18.34589020740345432913046742287, 19.0363921424993467696265881608
