L(s) = 1 | + (0.998 + 0.0475i)2-s + (0.995 + 0.0950i)4-s + (0.723 − 0.690i)5-s + (−0.981 − 0.189i)7-s + (0.989 + 0.142i)8-s + (0.755 − 0.654i)10-s + (−0.458 + 0.888i)11-s + (−0.981 + 0.189i)13-s + (−0.971 − 0.235i)14-s + (0.981 + 0.189i)16-s + (−0.909 − 0.415i)17-s + (0.909 − 0.415i)19-s + (0.786 − 0.618i)20-s + (−0.5 + 0.866i)22-s + (0.0475 − 0.998i)25-s + (−0.989 + 0.142i)26-s + ⋯ |
L(s) = 1 | + (0.998 + 0.0475i)2-s + (0.995 + 0.0950i)4-s + (0.723 − 0.690i)5-s + (−0.981 − 0.189i)7-s + (0.989 + 0.142i)8-s + (0.755 − 0.654i)10-s + (−0.458 + 0.888i)11-s + (−0.981 + 0.189i)13-s + (−0.971 − 0.235i)14-s + (0.981 + 0.189i)16-s + (−0.909 − 0.415i)17-s + (0.909 − 0.415i)19-s + (0.786 − 0.618i)20-s + (−0.5 + 0.866i)22-s + (0.0475 − 0.998i)25-s + (−0.989 + 0.142i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.203 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.203 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.659930252 - 2.040285112i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.659930252 - 2.040285112i\) |
\(L(1)\) |
\(\approx\) |
\(1.739767883 - 0.3530563131i\) |
\(L(1)\) |
\(\approx\) |
\(1.739767883 - 0.3530563131i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 23 | \( 1 \) |
| 29 | \( 1 \) |
good | 2 | \( 1 + (0.998 + 0.0475i)T \) |
| 5 | \( 1 + (0.723 - 0.690i)T \) |
| 7 | \( 1 + (-0.981 - 0.189i)T \) |
| 11 | \( 1 + (-0.458 + 0.888i)T \) |
| 13 | \( 1 + (-0.981 + 0.189i)T \) |
| 17 | \( 1 + (-0.909 - 0.415i)T \) |
| 19 | \( 1 + (0.909 - 0.415i)T \) |
| 31 | \( 1 + (0.618 - 0.786i)T \) |
| 37 | \( 1 + (-0.281 - 0.959i)T \) |
| 41 | \( 1 + (-0.690 - 0.723i)T \) |
| 43 | \( 1 + (0.618 + 0.786i)T \) |
| 47 | \( 1 + (-0.866 + 0.5i)T \) |
| 53 | \( 1 + (0.654 - 0.755i)T \) |
| 59 | \( 1 + (0.981 - 0.189i)T \) |
| 61 | \( 1 + (-0.371 - 0.928i)T \) |
| 67 | \( 1 + (-0.888 + 0.458i)T \) |
| 71 | \( 1 + (-0.841 - 0.540i)T \) |
| 73 | \( 1 + (0.909 - 0.415i)T \) |
| 79 | \( 1 + (0.945 + 0.327i)T \) |
| 83 | \( 1 + (-0.723 - 0.690i)T \) |
| 89 | \( 1 + (-0.989 + 0.142i)T \) |
| 97 | \( 1 + (-0.971 + 0.235i)T \) |
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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
−17.96824088050805497857716999573, −17.00145063935874594291166297535, −16.55382865792272880896913762958, −15.70022688851894182143433741507, −15.27572266878090370561137765704, −14.54952498832297942526525686949, −13.778730417910881755857313219459, −13.45151566471572339249748799707, −12.82197623926715659337858017695, −12.02367873993846817879212201168, −11.4723113856402168459859150402, −10.40736355818351139688233125043, −10.3109651986445471449673587576, −9.45664183103430310197347618471, −8.515448560060110312133037527007, −7.57337956649540850370979875431, −6.79527316552810604781144237899, −6.43104221882426700068596100316, −5.58314321208910990423034971329, −5.2218109656014293211777670637, −4.14273140469713317265650502405, −3.22875067882549326888197726558, −2.872570060734047990497567339920, −2.219056660722328824088684460710, −1.1855754818508603393926513443,
0.40198508923468491061396767081, 1.66090849328044021366693629161, 2.41212608620889193882786594539, 2.8787853407885364863943088197, 3.98347802723172086522954869165, 4.65572236044637117045425829962, 5.17812730420809793133398329916, 5.87844165894060354554846936874, 6.72751539502510794610188638612, 7.146600935376618374281601444948, 7.93008108743976109357365424500, 9.04761379730773067983283311476, 9.73135616123034428465839985100, 10.0715004953969567210224249548, 11.036146226787492526734258734248, 11.86479787350267005292620263571, 12.52095085160598021024636067219, 12.9040304300919935106996723053, 13.60625052684860826748732959176, 13.98186809422915651020569986425, 14.96106614501552579674493750721, 15.50600537694678756847489600862, 16.24301578288389937162837508831, 16.5824980725062258369222394240, 17.585053807703732941754274255