| L(s) = 1 | − 2-s + 4-s + 5-s + 7-s − 8-s − 10-s + 11-s + 13-s − 14-s + 16-s + 19-s + 20-s − 22-s + 23-s + 25-s − 26-s + 28-s − 29-s − 31-s − 32-s + 35-s + 37-s − 38-s − 40-s − 41-s − 43-s + 44-s + ⋯ |
| L(s) = 1 | − 2-s + 4-s + 5-s + 7-s − 8-s − 10-s + 11-s + 13-s − 14-s + 16-s + 19-s + 20-s − 22-s + 23-s + 25-s − 26-s + 28-s − 29-s − 31-s − 32-s + 35-s + 37-s − 38-s − 40-s − 41-s − 43-s + 44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.084468484\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.084468484\) |
| \(L(1)\) |
\(\approx\) |
\(1.147845333\) |
| \(L(1)\) |
\(\approx\) |
\(1.147845333\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 17 | \( 1 \) |
| 79 | \( 1 \) |
| good | 2 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| 31 | \( 1 - T \) |
| 37 | \( 1 + T \) |
| 41 | \( 1 - T \) |
| 43 | \( 1 - T \) |
| 47 | \( 1 + T \) |
| 53 | \( 1 + T \) |
| 59 | \( 1 + T \) |
| 61 | \( 1 + T \) |
| 67 | \( 1 + T \) |
| 71 | \( 1 - T \) |
| 73 | \( 1 - T \) |
| 83 | \( 1 - T \) |
| 89 | \( 1 - T \) |
| 97 | \( 1 - T \) |
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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
−18.386973307093903227795963825626, −17.85444983846759093302401694558, −17.13153247696397592187010649116, −16.73762318702579800068038939093, −15.99275712601706148347841300252, −14.88649985286978998185000052597, −14.68113051942151479481307923955, −13.70963699386366789650723459123, −13.07785751263058946340695614776, −12.024080044003489739622898934127, −11.3274217808175512408271703558, −10.9608895785557223814466036778, −10.05157014975687277584355083574, −9.38241998558975822650314666066, −8.80967899044090140404695289149, −8.25509619080849977246514192437, −7.17353249466548439407879905358, −6.8006436053400912539941468198, −5.67625859958234609028852211350, −5.43038505612439679982864124595, −4.10062397315718008326226105573, −3.20792123635871175187579455779, −2.23140790324331898705195431246, −1.42974823266999873665565187383, −1.068442598496684100854632562283,
1.068442598496684100854632562283, 1.42974823266999873665565187383, 2.23140790324331898705195431246, 3.20792123635871175187579455779, 4.10062397315718008326226105573, 5.43038505612439679982864124595, 5.67625859958234609028852211350, 6.8006436053400912539941468198, 7.17353249466548439407879905358, 8.25509619080849977246514192437, 8.80967899044090140404695289149, 9.38241998558975822650314666066, 10.05157014975687277584355083574, 10.9608895785557223814466036778, 11.3274217808175512408271703558, 12.024080044003489739622898934127, 13.07785751263058946340695614776, 13.70963699386366789650723459123, 14.68113051942151479481307923955, 14.88649985286978998185000052597, 15.99275712601706148347841300252, 16.73762318702579800068038939093, 17.13153247696397592187010649116, 17.85444983846759093302401694558, 18.386973307093903227795963825626
