L(s) = 1 | + (0.766 − 0.642i)2-s + (0.5 − 0.866i)3-s + (0.173 − 0.984i)4-s + i·5-s + (−0.173 − 0.984i)6-s + 7-s + (−0.5 − 0.866i)8-s + (−0.5 − 0.866i)9-s + (0.642 + 0.766i)10-s + (−0.642 + 0.766i)11-s + (−0.766 − 0.642i)12-s + (−0.5 + 0.866i)13-s + (0.766 − 0.642i)14-s + (0.866 + 0.5i)15-s + (−0.939 − 0.342i)16-s + (−0.173 − 0.984i)17-s + ⋯ |
L(s) = 1 | + (0.766 − 0.642i)2-s + (0.5 − 0.866i)3-s + (0.173 − 0.984i)4-s + i·5-s + (−0.173 − 0.984i)6-s + 7-s + (−0.5 − 0.866i)8-s + (−0.5 − 0.866i)9-s + (0.642 + 0.766i)10-s + (−0.642 + 0.766i)11-s + (−0.766 − 0.642i)12-s + (−0.5 + 0.866i)13-s + (0.766 − 0.642i)14-s + (0.866 + 0.5i)15-s + (−0.939 − 0.342i)16-s + (−0.173 − 0.984i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.629 + 0.777i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.629 + 0.777i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.268745439 + 0.6051274355i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.268745439 + 0.6051274355i\) |
\(L(1)\) |
\(\approx\) |
\(1.390941169 - 0.5637277998i\) |
\(L(1)\) |
\(\approx\) |
\(1.390941169 - 0.5637277998i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 37 | \( 1 \) |
| 109 | \( 1 \) |
good | 2 | \( 1 + (0.766 - 0.642i)T \) |
| 3 | \( 1 + (0.5 - 0.866i)T \) |
| 5 | \( 1 + iT \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + (-0.642 + 0.766i)T \) |
| 13 | \( 1 + (-0.5 + 0.866i)T \) |
| 17 | \( 1 + (-0.173 - 0.984i)T \) |
| 19 | \( 1 + (-0.939 + 0.342i)T \) |
| 23 | \( 1 + (0.5 + 0.866i)T \) |
| 29 | \( 1 + (-0.984 + 0.173i)T \) |
| 31 | \( 1 + (-0.642 + 0.766i)T \) |
| 41 | \( 1 + (0.342 + 0.939i)T \) |
| 43 | \( 1 + (-0.866 + 0.5i)T \) |
| 47 | \( 1 + (-0.642 + 0.766i)T \) |
| 53 | \( 1 + (0.984 + 0.173i)T \) |
| 59 | \( 1 + (-0.5 + 0.866i)T \) |
| 61 | \( 1 + (0.866 - 0.5i)T \) |
| 67 | \( 1 + (-0.984 + 0.173i)T \) |
| 71 | \( 1 + (-0.766 - 0.642i)T \) |
| 73 | \( 1 + (0.939 - 0.342i)T \) |
| 79 | \( 1 + (0.173 + 0.984i)T \) |
| 83 | \( 1 + (-0.766 - 0.642i)T \) |
| 89 | \( 1 + (0.642 - 0.766i)T \) |
| 97 | \( 1 + (-0.984 + 0.173i)T \) |
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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
−18.15728442531377868236143544563, −17.21370718990316198245398052470, −16.91457284321543813906425134886, −16.3184831458833373981706691367, −15.386349972434918576295848301013, −15.06192461093503929901525676534, −14.52776919699549563223060557339, −13.56045086531044639494222248396, −13.10343859923539807296799075246, −12.49584887375028062777823764818, −11.46366684170138096597383314346, −10.88411880314073881932543889577, −10.15814696701519606729524456761, −8.908728412339899320868899798443, −8.5617481737497561668046576531, −8.04572472608973172186670497651, −7.38416684873305911861210238433, −6.04896865129482645447479808428, −5.33211835185363190441252463041, −5.023150747915253817315138258301, −4.15794169377678031089437820212, −3.67649664128566822153514864293, −2.54295584267259795476818055438, −1.94649934176989559650428478455, −0.23103419569836986261314188844,
1.438075506661343155187936193154, 1.96740770143049000116049299912, 2.57783951890891116618016828904, 3.30574720894112185854615323598, 4.25271606548721089047892782289, 4.99060639630261741455339705135, 5.82827844433938209864824165387, 6.76562079591086990002986043049, 7.23416003795347449518074043948, 7.83210361136651042061973145021, 8.993629894982190640712246654219, 9.637579631427478673616330454057, 10.48676660802929108327961756791, 11.32543179279386528612392995407, 11.61166929281494934479638412496, 12.4466044215051613944491651944, 13.16151106655580795181879311907, 13.80625168712104277847749555911, 14.43274146465349680520708356753, 14.90047130886458952362487372506, 15.27959583572050025620215888802, 16.49742814196932879852298982721, 17.644512144904863140165921108188, 18.11149530711832972466705643119, 18.620752951651744697729315707768