L(s) = 1 | + i·2-s − 4-s − i·7-s − i·8-s + 11-s + i·13-s + 14-s + 16-s − i·17-s − 19-s + i·22-s + i·23-s − 26-s + i·28-s + 29-s + ⋯ |
L(s) = 1 | + i·2-s − 4-s − i·7-s − i·8-s + 11-s + i·13-s + 14-s + 16-s − i·17-s − 19-s + i·22-s + i·23-s − 26-s + i·28-s + 29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1005 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1005 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.697442509 + 0.4822078449i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.697442509 + 0.4822078449i\) |
\(L(1)\) |
\(\approx\) |
\(0.9471799992 + 0.3523950139i\) |
\(L(1)\) |
\(\approx\) |
\(0.9471799992 + 0.3523950139i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 67 | \( 1 \) |
good | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 \) |
| 17 | \( 1 \) |
| 19 | \( 1 - iT \) |
| 23 | \( 1 - iT \) |
| 29 | \( 1 \) |
| 31 | \( 1 \) |
| 37 | \( 1 + T \) |
| 41 | \( 1 \) |
| 43 | \( 1 + iT \) |
| 47 | \( 1 + T \) |
| 53 | \( 1 \) |
| 59 | \( 1 + T \) |
| 61 | \( 1 - iT \) |
| 71 | \( 1 - T \) |
| 73 | \( 1 \) |
| 79 | \( 1 \) |
| 83 | \( 1 + iT \) |
| 89 | \( 1 + iT \) |
| 97 | \( 1 \) |
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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
−21.45338178088138519437836637293, −20.612182657040857715145587196362, −19.70266878992783359401064691174, −19.24134316390995508623887864406, −18.3871577375208726193300130357, −17.61954323654189700804856355157, −16.98306312794864229011759646118, −15.786232395079977244878962560, −14.73096911760217287924643778236, −14.41617575803320049090540066908, −13.04671052044270773971428478736, −12.53535017819742704845875111109, −11.938808463254354206593152479779, −10.8464554953038193645137057034, −10.368824090325610637769062966545, −9.14536598363157565698701538550, −8.73906943838041984967572311589, −7.839272122565115244855228384363, −6.3253188794350743361068879443, −5.65191446525081684881951725066, −4.53505346271256707353819996100, −3.71291884560768205824358157254, −2.67150725397171957102370154274, −1.88536672050819502894094165308, −0.71933236025749455050358444244,
0.56020677508324765590137507361, 1.640539681324770953689657737795, 3.37670412206671324259578189721, 4.22878716447725414223829209721, 4.85199948687617871920592938873, 6.14080991556159539331944894594, 6.830670393033766163628018353492, 7.41928025273554803074291674513, 8.47807685200846114583614825060, 9.31156147065024919511788588683, 9.95201186343613274701528867575, 11.113808076543062992293600652094, 11.98238513815467564563220154759, 13.10197900198707109483154215107, 13.82530602666650713892610630479, 14.36980670362486687423937020132, 15.153478148776811731879378207382, 16.33112304179245669405186238804, 16.59781925054948247080492008572, 17.46779578858848385500863302560, 18.11429914800389395154141500955, 19.26958105762745217936837914853, 19.65028592328385870525422581522, 20.86800046839649754722832666533, 21.683989822914552727933512879538