L(s) = 1 | + (−0.797 − 0.603i)2-s + (0.272 + 0.962i)4-s + (−0.710 − 0.703i)5-s + (0.179 + 0.983i)7-s + (0.362 − 0.931i)8-s + (0.142 + 0.989i)10-s + (0.830 − 0.556i)13-s + (0.449 − 0.893i)14-s + (−0.851 + 0.524i)16-s + (0.736 − 0.676i)17-s + (−0.198 − 0.980i)19-s + (0.483 − 0.875i)20-s + (0.723 − 0.690i)23-s + (0.00951 + 0.999i)25-s + (−0.998 − 0.0570i)26-s + ⋯ |
L(s) = 1 | + (−0.797 − 0.603i)2-s + (0.272 + 0.962i)4-s + (−0.710 − 0.703i)5-s + (0.179 + 0.983i)7-s + (0.362 − 0.931i)8-s + (0.142 + 0.989i)10-s + (0.830 − 0.556i)13-s + (0.449 − 0.893i)14-s + (−0.851 + 0.524i)16-s + (0.736 − 0.676i)17-s + (−0.198 − 0.980i)19-s + (0.483 − 0.875i)20-s + (0.723 − 0.690i)23-s + (0.00951 + 0.999i)25-s + (−0.998 − 0.0570i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.600 - 0.799i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.600 - 0.799i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4786711238 - 0.9585071925i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4786711238 - 0.9585071925i\) |
\(L(1)\) |
\(\approx\) |
\(0.6576367921 - 0.2767230374i\) |
\(L(1)\) |
\(\approx\) |
\(0.6576367921 - 0.2767230374i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-0.797 - 0.603i)T \) |
| 5 | \( 1 + (-0.710 - 0.703i)T \) |
| 7 | \( 1 + (0.179 + 0.983i)T \) |
| 13 | \( 1 + (0.830 - 0.556i)T \) |
| 17 | \( 1 + (0.736 - 0.676i)T \) |
| 19 | \( 1 + (-0.198 - 0.980i)T \) |
| 23 | \( 1 + (0.723 - 0.690i)T \) |
| 29 | \( 1 + (-0.861 - 0.508i)T \) |
| 31 | \( 1 + (0.640 - 0.768i)T \) |
| 37 | \( 1 + (-0.466 + 0.884i)T \) |
| 41 | \( 1 + (0.683 - 0.730i)T \) |
| 43 | \( 1 + (0.888 + 0.458i)T \) |
| 47 | \( 1 + (-0.905 + 0.424i)T \) |
| 53 | \( 1 + (-0.0285 + 0.999i)T \) |
| 59 | \( 1 + (-0.290 + 0.956i)T \) |
| 61 | \( 1 + (-0.123 - 0.992i)T \) |
| 67 | \( 1 + (0.981 - 0.189i)T \) |
| 71 | \( 1 + (0.993 + 0.113i)T \) |
| 73 | \( 1 + (-0.610 - 0.791i)T \) |
| 79 | \( 1 + (0.935 + 0.353i)T \) |
| 83 | \( 1 + (0.432 - 0.901i)T \) |
| 89 | \( 1 + (0.415 + 0.909i)T \) |
| 97 | \( 1 + (-0.710 + 0.703i)T \) |
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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
−21.28022050507378766417338186230, −20.63899563200951325087331606110, −19.57440403547932270907532832760, −19.20674633416587078239662625185, −18.407230828463228986127193401825, −17.64811674852095066711958934743, −16.761344574590210986426355838072, −16.21102853327381357580177876116, −15.38655717005422523978380429791, −14.43327700682021903477292156173, −14.16199138370010253916705641885, −12.92878797570531495690728508007, −11.68311859547413108146127492162, −10.93221954496304849779854598639, −10.45148933044148690807204649517, −9.53009676007247792813427899989, −8.41719529523212951808610230140, −7.80770879233304986048848585814, −7.051552527792641519786424437896, −6.37766367904111003391323489367, −5.33983480680323365666655947230, −4.080720619638700651405278592018, −3.38327543106721558393570849666, −1.80447322121933528435719497249, −0.9208578244184598291644712084,
0.39043917524746764654955239596, 1.19140602798548759297393430408, 2.46900786068657443427827675741, 3.25395242511585993742308516862, 4.34002169308624210228664995540, 5.26941661659879739200739989086, 6.42568479970897635028184900289, 7.63549112064343407729726981908, 8.1893380167935723260819846209, 9.01761615219191239724827692337, 9.502363910177359604981809782434, 10.82207914632229628807875474175, 11.38321666628131447041085811814, 12.16719979002831109925351232585, 12.77415388405544277954118141775, 13.58894112801808453342967418334, 15.0849015227085276460299532233, 15.6272170497438292311104415415, 16.36318661635080668270880532506, 17.18008156531052880761778024877, 17.98492408494164493718336181785, 18.890935741447464209701222219966, 19.157959968750714355246411999492, 20.34882266641180285072835703471, 20.73055382126785628254289054710