L(s) = 1 | + (0.179 − 0.983i)2-s + (−0.669 + 0.743i)3-s + (−0.935 − 0.353i)4-s + (0.290 − 0.956i)5-s + (0.610 + 0.791i)6-s + (−0.516 + 0.856i)8-s + (−0.104 − 0.994i)9-s + (−0.888 − 0.458i)10-s + (0.888 − 0.458i)12-s + (−0.998 + 0.0570i)13-s + (0.516 + 0.856i)15-s + (0.749 + 0.662i)16-s + (0.999 − 0.0380i)17-s + (−0.997 − 0.0760i)18-s + (0.345 − 0.938i)19-s + (−0.610 + 0.791i)20-s + ⋯ |
L(s) = 1 | + (0.179 − 0.983i)2-s + (−0.669 + 0.743i)3-s + (−0.935 − 0.353i)4-s + (0.290 − 0.956i)5-s + (0.610 + 0.791i)6-s + (−0.516 + 0.856i)8-s + (−0.104 − 0.994i)9-s + (−0.888 − 0.458i)10-s + (0.888 − 0.458i)12-s + (−0.998 + 0.0570i)13-s + (0.516 + 0.856i)15-s + (0.749 + 0.662i)16-s + (0.999 − 0.0380i)17-s + (−0.997 − 0.0760i)18-s + (0.345 − 0.938i)19-s + (−0.610 + 0.791i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.729 + 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.729 + 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.1266930497 - 0.3202555484i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.1266930497 - 0.3202555484i\) |
\(L(1)\) |
\(\approx\) |
\(0.5795489174 - 0.3772706884i\) |
\(L(1)\) |
\(\approx\) |
\(0.5795489174 - 0.3772706884i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (0.179 - 0.983i)T \) |
| 3 | \( 1 + (-0.669 + 0.743i)T \) |
| 5 | \( 1 + (0.290 - 0.956i)T \) |
| 13 | \( 1 + (-0.998 + 0.0570i)T \) |
| 17 | \( 1 + (0.999 - 0.0380i)T \) |
| 19 | \( 1 + (0.345 - 0.938i)T \) |
| 23 | \( 1 + (-0.995 - 0.0950i)T \) |
| 29 | \( 1 + (-0.897 - 0.441i)T \) |
| 31 | \( 1 + (-0.988 + 0.151i)T \) |
| 37 | \( 1 + (0.964 + 0.263i)T \) |
| 41 | \( 1 + (-0.0285 - 0.999i)T \) |
| 43 | \( 1 + (0.654 + 0.755i)T \) |
| 47 | \( 1 + (-0.997 + 0.0760i)T \) |
| 53 | \( 1 + (0.749 - 0.662i)T \) |
| 59 | \( 1 + (-0.879 + 0.475i)T \) |
| 61 | \( 1 + (-0.761 + 0.647i)T \) |
| 67 | \( 1 + (0.235 + 0.971i)T \) |
| 71 | \( 1 + (0.696 - 0.717i)T \) |
| 73 | \( 1 + (-0.595 - 0.803i)T \) |
| 79 | \( 1 + (-0.820 - 0.572i)T \) |
| 83 | \( 1 + (-0.870 + 0.491i)T \) |
| 89 | \( 1 + (-0.928 - 0.371i)T \) |
| 97 | \( 1 + (-0.974 - 0.226i)T \) |
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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
−22.8725679907641771879893868038, −21.96147654255481947676541126383, −21.6116674851948195991067961739, −20.041072784803496367716258344856, −18.94212132195624791176084653202, −18.41080051672426065045635059145, −17.86768109017097252254103050503, −16.86501173666633595121044224090, −16.499930404353310504966969412779, −15.28632197257590391803948634917, −14.36980945844631978853747714430, −14.04078110923954283882403349758, −12.87793282004366926749001786904, −12.274415668426249942793241514701, −11.301236077821225276745788625205, −10.17017058128597494709700043721, −9.53033397576686644178511873251, −7.950266932151385858362964025, −7.55386144979585392613303839284, −6.75057001796751809114944142341, −5.81181388430537566885303983897, −5.40562969917957210911376234344, −4.045235991441284138036312965386, −2.90151021736680337574818998017, −1.61569821497007750678169291038,
0.166128604029968040879536302353, 1.35950144668190550196729979025, 2.60514315283449932124876924434, 3.80470576924280853563815779610, 4.59329698140245501299507575449, 5.320818456341056112457544479551, 5.96568957172410666832660348585, 7.58556942520500323584082539256, 8.80568513252759122584930451828, 9.59436069945258072254206707547, 9.97495281664412493213251227064, 11.06062534838609475538211306186, 11.87447896303278906928195761853, 12.42397429493459649075512026168, 13.255059180641875415862244856720, 14.298538069430894165468011053340, 15.06104360001269087436753777557, 16.211857219951689225219540388347, 16.87815362462452325302727226872, 17.629029807056809560099062661616, 18.335256842483531744907121764098, 19.570612732358606752707343735419, 20.16210079359420156019041554304, 20.93733738009963609039291240189, 21.56235867706013410392097199308