L(s) = 1 | + (0.900 − 0.433i)2-s + (0.900 + 0.433i)3-s + (0.623 − 0.781i)4-s + (0.222 − 0.974i)5-s + 6-s − 7-s + (0.222 − 0.974i)8-s + (0.623 + 0.781i)9-s + (−0.222 − 0.974i)10-s + (0.623 + 0.781i)11-s + (0.900 − 0.433i)12-s + (−0.222 + 0.974i)13-s + (−0.900 + 0.433i)14-s + (0.623 − 0.781i)15-s + (−0.222 − 0.974i)16-s + (−0.222 − 0.974i)17-s + ⋯ |
L(s) = 1 | + (0.900 − 0.433i)2-s + (0.900 + 0.433i)3-s + (0.623 − 0.781i)4-s + (0.222 − 0.974i)5-s + 6-s − 7-s + (0.222 − 0.974i)8-s + (0.623 + 0.781i)9-s + (−0.222 − 0.974i)10-s + (0.623 + 0.781i)11-s + (0.900 − 0.433i)12-s + (−0.222 + 0.974i)13-s + (−0.900 + 0.433i)14-s + (0.623 − 0.781i)15-s + (−0.222 − 0.974i)16-s + (−0.222 − 0.974i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.764 - 0.644i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.764 - 0.644i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.854530500 - 1.042566372i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.854530500 - 1.042566372i\) |
\(L(1)\) |
\(\approx\) |
\(2.085238254 - 0.5359162614i\) |
\(L(1)\) |
\(\approx\) |
\(2.085238254 - 0.5359162614i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 43 | \( 1 \) |
good | 2 | \( 1 + (0.900 - 0.433i)T \) |
| 3 | \( 1 + (0.900 + 0.433i)T \) |
| 5 | \( 1 + (0.222 - 0.974i)T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + (0.623 + 0.781i)T \) |
| 13 | \( 1 + (-0.222 + 0.974i)T \) |
| 17 | \( 1 + (-0.222 - 0.974i)T \) |
| 19 | \( 1 + (-0.623 + 0.781i)T \) |
| 23 | \( 1 + (0.623 + 0.781i)T \) |
| 29 | \( 1 + (0.900 - 0.433i)T \) |
| 31 | \( 1 + (-0.900 + 0.433i)T \) |
| 37 | \( 1 - T \) |
| 41 | \( 1 + (-0.900 + 0.433i)T \) |
| 47 | \( 1 + (0.623 - 0.781i)T \) |
| 53 | \( 1 + (-0.222 - 0.974i)T \) |
| 59 | \( 1 + (-0.222 - 0.974i)T \) |
| 61 | \( 1 + (0.900 + 0.433i)T \) |
| 67 | \( 1 + (0.623 - 0.781i)T \) |
| 71 | \( 1 + (-0.623 + 0.781i)T \) |
| 73 | \( 1 + (0.222 - 0.974i)T \) |
| 79 | \( 1 + T \) |
| 83 | \( 1 + (-0.900 - 0.433i)T \) |
| 89 | \( 1 + (0.900 + 0.433i)T \) |
| 97 | \( 1 + (0.623 + 0.781i)T \) |
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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
−34.70656099326627358234079449553, −32.80096279902100596008985127917, −32.338369667982740413921288210137, −30.96018633354183085239523890556, −30.04818721333249977320462150, −29.32319450516873887648972963791, −26.82637976591400406806571543603, −25.8529839098270901761993117270, −25.12205904384124783100681287190, −23.82413347486548473837441290115, −22.47843029077344574949413736380, −21.610653028644928559796269045112, −19.98124984108376926619420220458, −18.98043131557391233225787332149, −17.29907724035354228504377961501, −15.56303457411751382369023405759, −14.64306550674612405258920083662, −13.51467506651064694727770087223, −12.53776141115387384298445784071, −10.624637400517962226776630706, −8.68066289124399190278859242854, −7.06517941065066446739017838451, −6.17262425776981166280704721998, −3.64295700763236810088040677086, −2.66104952663855579404469995350,
1.93292510242362461435598072612, 3.70143206259274239207471155869, 4.91655164304745600317631356842, 6.870303265390711268751065935971, 9.12783195867501763816479724919, 9.967497830134490270808167796365, 12.03905525112830027898988674780, 13.16584963329658233927714254799, 14.20357068817370693925387890603, 15.59586281285417889450359737043, 16.62134024463874222709080230089, 19.13676703649877165265712240422, 19.99552803801580546188923811777, 20.95912398115675883027735066942, 22.015588638798288249501301793743, 23.36799306108290632357520413851, 24.890135897734906352973187725781, 25.48743243302504162766592752498, 27.32064242244484844160343794524, 28.58839477390057686966903383814, 29.5902863241859644848128831010, 31.15652302938014421790881058635, 31.78603494443456812392215569215, 32.77412908802359701834632586773, 33.53077994353415543493500950739