L(s) = 1 | + (−0.909 + 0.415i)2-s + (0.654 − 0.755i)4-s + (0.841 + 0.540i)5-s + (−0.142 − 0.989i)7-s + (−0.281 + 0.959i)8-s + (−0.989 − 0.142i)10-s + (0.909 + 0.415i)11-s + (0.142 − 0.989i)13-s + (0.540 + 0.841i)14-s + (−0.142 − 0.989i)16-s + (0.755 − 0.654i)17-s + (0.755 + 0.654i)19-s + (0.959 − 0.281i)20-s − 22-s + (0.415 + 0.909i)25-s + (0.281 + 0.959i)26-s + ⋯ |
L(s) = 1 | + (−0.909 + 0.415i)2-s + (0.654 − 0.755i)4-s + (0.841 + 0.540i)5-s + (−0.142 − 0.989i)7-s + (−0.281 + 0.959i)8-s + (−0.989 − 0.142i)10-s + (0.909 + 0.415i)11-s + (0.142 − 0.989i)13-s + (0.540 + 0.841i)14-s + (−0.142 − 0.989i)16-s + (0.755 − 0.654i)17-s + (0.755 + 0.654i)19-s + (0.959 − 0.281i)20-s − 22-s + (0.415 + 0.909i)25-s + (0.281 + 0.959i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.961 + 0.275i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.961 + 0.275i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.446519894 + 0.2028359003i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.446519894 + 0.2028359003i\) |
\(L(1)\) |
\(\approx\) |
\(0.9479550489 + 0.1175974141i\) |
\(L(1)\) |
\(\approx\) |
\(0.9479550489 + 0.1175974141i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 23 | \( 1 \) |
| 29 | \( 1 \) |
good | 2 | \( 1 + (-0.909 + 0.415i)T \) |
| 5 | \( 1 + (0.841 + 0.540i)T \) |
| 7 | \( 1 + (-0.142 - 0.989i)T \) |
| 11 | \( 1 + (0.909 + 0.415i)T \) |
| 13 | \( 1 + (0.142 - 0.989i)T \) |
| 17 | \( 1 + (0.755 - 0.654i)T \) |
| 19 | \( 1 + (0.755 + 0.654i)T \) |
| 31 | \( 1 + (-0.281 + 0.959i)T \) |
| 37 | \( 1 + (0.540 + 0.841i)T \) |
| 41 | \( 1 + (0.540 - 0.841i)T \) |
| 43 | \( 1 + (0.281 + 0.959i)T \) |
| 47 | \( 1 - iT \) |
| 53 | \( 1 + (0.142 + 0.989i)T \) |
| 59 | \( 1 + (0.142 - 0.989i)T \) |
| 61 | \( 1 + (-0.281 + 0.959i)T \) |
| 67 | \( 1 + (-0.415 - 0.909i)T \) |
| 71 | \( 1 + (0.415 + 0.909i)T \) |
| 73 | \( 1 + (-0.755 - 0.654i)T \) |
| 79 | \( 1 + (0.989 + 0.142i)T \) |
| 83 | \( 1 + (-0.841 + 0.540i)T \) |
| 89 | \( 1 + (-0.281 - 0.959i)T \) |
| 97 | \( 1 + (-0.540 + 0.841i)T \) |
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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
−19.677439903485314273808042726613, −19.23752915702449442582595978871, −18.42761431289621814559737723925, −17.850495742842327355217102242924, −17.0086436372707946541495461934, −16.513003118599317229245499316258, −15.87431482247226172072177593411, −14.82783546448751451367452757500, −14.03322240112619614357519795350, −13.0792530884093398311979772706, −12.41332095127237484624317929532, −11.68365150016377277422088316448, −11.12220830112818784909578521674, −9.954361930519040537969702084521, −9.34296986063852916713631403043, −8.97828956749257330271813737295, −8.20582398649741432258878041688, −7.15468309299598500222155418001, −6.18394543855617694391049921428, −5.7573703432909394293798803527, −4.46820093702334547671046350615, −3.47592875886885229845560561284, −2.478027382114207122512787906830, −1.7485595738738790274348310293, −0.922918104717191117216303859844,
0.94401317149519293157865304213, 1.54030095730719661372912517509, 2.78232796187051591788839075714, 3.5564456831301235597167423475, 4.93035036988621480656623055286, 5.76830631843227092772883381913, 6.48411188710005046362193127198, 7.2787305553210894074952257193, 7.73194909268664049082658140185, 8.874711225631323670146083565252, 9.723857794326062943493072866452, 10.10007862913591281215009609766, 10.775308919903020589529063565092, 11.6297921797929807692027202250, 12.588942516371146693154532962660, 13.712107895246330554000054451374, 14.226347533792800675961454795735, 14.81616192810012829247266312133, 15.75673259240846537412914952946, 16.637895384758375035432535899485, 17.04945000651616823530944722245, 17.89738499467605520189536549370, 18.23413033784092654643201006321, 19.176035974202793302522587557309, 19.97144891457763955633675247463