L(s) = 1 | + (0.904 − 0.425i)2-s + (0.844 − 0.535i)3-s + (0.637 − 0.770i)4-s + (0.535 − 0.844i)6-s + (0.587 − 0.809i)7-s + (0.248 − 0.968i)8-s + (0.425 − 0.904i)9-s + (0.125 − 0.992i)12-s + (0.904 + 0.425i)13-s + (0.187 − 0.982i)14-s + (−0.187 − 0.982i)16-s + (−0.248 + 0.968i)17-s − i·18-s + (−0.0627 − 0.998i)19-s + (0.0627 − 0.998i)21-s + ⋯ |
L(s) = 1 | + (0.904 − 0.425i)2-s + (0.844 − 0.535i)3-s + (0.637 − 0.770i)4-s + (0.535 − 0.844i)6-s + (0.587 − 0.809i)7-s + (0.248 − 0.968i)8-s + (0.425 − 0.904i)9-s + (0.125 − 0.992i)12-s + (0.904 + 0.425i)13-s + (0.187 − 0.982i)14-s + (−0.187 − 0.982i)16-s + (−0.248 + 0.968i)17-s − i·18-s + (−0.0627 − 0.998i)19-s + (0.0627 − 0.998i)21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.918 - 0.395i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.918 - 0.395i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.230470262 - 5.968421245i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.230470262 - 5.968421245i\) |
\(L(1)\) |
\(\approx\) |
\(1.969495072 - 1.711247902i\) |
\(L(1)\) |
\(\approx\) |
\(1.969495072 - 1.711247902i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (0.904 - 0.425i)T \) |
| 3 | \( 1 + (0.844 - 0.535i)T \) |
| 7 | \( 1 + (0.587 - 0.809i)T \) |
| 13 | \( 1 + (0.904 + 0.425i)T \) |
| 17 | \( 1 + (-0.248 + 0.968i)T \) |
| 19 | \( 1 + (-0.0627 - 0.998i)T \) |
| 23 | \( 1 + (-0.904 + 0.425i)T \) |
| 29 | \( 1 + (-0.535 - 0.844i)T \) |
| 31 | \( 1 + (0.535 - 0.844i)T \) |
| 37 | \( 1 + (0.481 + 0.876i)T \) |
| 41 | \( 1 + (-0.992 - 0.125i)T \) |
| 43 | \( 1 + (-0.587 + 0.809i)T \) |
| 47 | \( 1 + (-0.770 - 0.637i)T \) |
| 53 | \( 1 + (0.844 - 0.535i)T \) |
| 59 | \( 1 + (0.425 - 0.904i)T \) |
| 61 | \( 1 + (-0.187 + 0.982i)T \) |
| 67 | \( 1 + (0.770 - 0.637i)T \) |
| 71 | \( 1 + (0.0627 - 0.998i)T \) |
| 73 | \( 1 + (0.982 + 0.187i)T \) |
| 79 | \( 1 + (0.637 - 0.770i)T \) |
| 83 | \( 1 + (-0.770 + 0.637i)T \) |
| 89 | \( 1 + (0.992 - 0.125i)T \) |
| 97 | \( 1 + (0.998 + 0.0627i)T \) |
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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
−21.06635885714230792680861984617, −20.43581603322836882421887306603, −19.88781386473319298076636636567, −18.56175923178195836298326425358, −18.11376222690597910208757758663, −16.87761765786045206752287511500, −15.96527906848809776280852665085, −15.70763438149211842420769822623, −14.73589924302529693869009345835, −14.27965885053081703555573661822, −13.559728131703718890858160403870, −12.698563758192609670375927967492, −11.8865290709900813365908880561, −11.042522948447198788595880277496, −10.19868523681382281325359588954, −9.00101421777061536128903816419, −8.369331653914780160800467786212, −7.773656128977976619215010386477, −6.71017753046346965750594754969, −5.64043404227475508898750094371, −5.04857684149258144044479899922, −4.09140597531735233595499462490, −3.3339868739816145301238989253, −2.46930281516275859297061972220, −1.619395194098296737759979705291,
0.656580172770290664371928426372, 1.625309484968959487035935449486, 2.23979232550171417124043535108, 3.50530909666810311928373759849, 3.98622224508041389330836531391, 4.86842596914582939861690257606, 6.20654083314219161338175446658, 6.687826370533627110810571914786, 7.74314355244548843866666159107, 8.407092809482208045475317741998, 9.560289699272650895836368893956, 10.312522424895314736369768634497, 11.34347241901868218617709862965, 11.77985838624108225653126950951, 13.03909680911556272129861138603, 13.445516550299976270758743890161, 13.91964995751741864442377004481, 14.95206252319891560934033244039, 15.26978339368658301023519540856, 16.38182331580719279686633672860, 17.39415577452195234404343917938, 18.29287133359557403268951378939, 19.04127445014321127471771569514, 19.88346273484165637765957426462, 20.215768961641521603822600794173