L(s) = 1 | + (0.833 + 0.552i)2-s + (0.390 + 0.920i)4-s + (−0.00679 − 0.999i)5-s + (−0.182 + 0.983i)8-s + (0.546 − 0.837i)10-s + (−0.511 + 0.859i)11-s + (0.979 + 0.202i)13-s + (−0.694 + 0.719i)16-s + (0.601 + 0.798i)17-s + (0.888 − 0.458i)19-s + (0.917 − 0.396i)20-s + (−0.900 + 0.433i)22-s + (−0.999 + 0.0135i)25-s + (0.704 + 0.709i)26-s + (0.992 − 0.122i)29-s + ⋯ |
L(s) = 1 | + (0.833 + 0.552i)2-s + (0.390 + 0.920i)4-s + (−0.00679 − 0.999i)5-s + (−0.182 + 0.983i)8-s + (0.546 − 0.837i)10-s + (−0.511 + 0.859i)11-s + (0.979 + 0.202i)13-s + (−0.694 + 0.719i)16-s + (0.601 + 0.798i)17-s + (0.888 − 0.458i)19-s + (0.917 − 0.396i)20-s + (−0.900 + 0.433i)22-s + (−0.999 + 0.0135i)25-s + (0.704 + 0.709i)26-s + (0.992 − 0.122i)29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.212 + 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.212 + 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.339012294 + 1.885566386i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.339012294 + 1.885566386i\) |
\(L(1)\) |
\(\approx\) |
\(1.655841240 + 0.6309526347i\) |
\(L(1)\) |
\(\approx\) |
\(1.655841240 + 0.6309526347i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 23 | \( 1 \) |
good | 2 | \( 1 + (0.833 + 0.552i)T \) |
| 5 | \( 1 + (-0.00679 - 0.999i)T \) |
| 11 | \( 1 + (-0.511 + 0.859i)T \) |
| 13 | \( 1 + (0.979 + 0.202i)T \) |
| 17 | \( 1 + (0.601 + 0.798i)T \) |
| 19 | \( 1 + (0.888 - 0.458i)T \) |
| 29 | \( 1 + (0.992 - 0.122i)T \) |
| 31 | \( 1 + (0.327 - 0.945i)T \) |
| 37 | \( 1 + (-0.675 - 0.737i)T \) |
| 41 | \( 1 + (-0.862 + 0.505i)T \) |
| 43 | \( 1 + (0.182 + 0.983i)T \) |
| 47 | \( 1 + (0.955 + 0.294i)T \) |
| 53 | \( 1 + (0.169 - 0.985i)T \) |
| 59 | \( 1 + (-0.546 + 0.837i)T \) |
| 61 | \( 1 + (0.966 + 0.255i)T \) |
| 67 | \( 1 + (0.235 + 0.971i)T \) |
| 71 | \( 1 + (-0.986 + 0.162i)T \) |
| 73 | \( 1 + (-0.963 - 0.268i)T \) |
| 79 | \( 1 + (0.580 + 0.814i)T \) |
| 83 | \( 1 + (0.996 + 0.0815i)T \) |
| 89 | \( 1 + (0.994 - 0.108i)T \) |
| 97 | \( 1 + (0.142 - 0.989i)T \) |
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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
−18.74726995867354592344367609444, −18.34963264751907866122034199738, −17.46258353998813726325853600134, −16.14017891969566407393800263942, −15.88677495654109234473176423969, −15.145275214998391434705195659911, −14.22351597570098222546590272363, −13.76191954117279848813154871119, −13.44623613744475636085020771359, −12.03077465414224460750520608751, −11.96554626353488167306193657095, −10.80236024496982562094530125146, −10.59225162865579277814184014178, −9.83889848114497159051248855925, −8.83935331505077717056337681660, −7.90074086845568428287467492175, −7.03363028810602236467429600004, −6.35979495966184325315821452103, −5.601172788292076412405796135713, −5.04488225189697230168920476603, −3.82460746753882196771469558451, −3.21489669115013057761760573816, −2.81981132943643449951787806366, −1.68085418059604513877588355975, −0.727968275787152330927734630178,
1.07762170900489842814297619172, 2.01331585530306694798550143765, 3.00202801082167409966711369139, 3.95275147574972122317578889033, 4.49939968463531567210402616931, 5.31200437797386399687674416622, 5.85497422203314551382451827809, 6.72646441861863662023024298953, 7.60687360626016474454016642989, 8.18639326523291267907811485268, 8.85813026063771187278232041084, 9.740322996773805096410187285640, 10.624842284612750372571953023409, 11.6630801600149890948723746183, 12.083026466447981004334030955771, 12.94042997602994352102241442083, 13.29990861104608242324722613899, 14.03635160314524647522389558873, 14.891334146402975515332305067340, 15.59390634395415197006439608686, 16.09301901169266035886261910748, 16.67681652449374797684156258281, 17.57391747515635928511788885171, 17.88435595212258028913807095637, 19.03449437652481029691076216395