L(s) = 1 | + 3-s + 5-s + 9-s + 11-s + 15-s − 17-s − 19-s − 23-s + 25-s + 27-s + 29-s − 31-s + 33-s − 37-s + 41-s − 43-s + 45-s − 47-s − 51-s + 53-s + 55-s − 57-s − 59-s − 61-s + 67-s − 69-s + 71-s + ⋯ |
L(s) = 1 | + 3-s + 5-s + 9-s + 11-s + 15-s − 17-s − 19-s − 23-s + 25-s + 27-s + 29-s − 31-s + 33-s − 37-s + 41-s − 43-s + 45-s − 47-s − 51-s + 53-s + 55-s − 57-s − 59-s − 61-s + 67-s − 69-s + 71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 364 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 364 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.184045669\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.184045669\) |
\(L(1)\) |
\(\approx\) |
\(1.688686762\) |
\(L(1)\) |
\(\approx\) |
\(1.688686762\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 \) |
good | 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 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 | \( 1 + T \) |
| 79 | \( 1 - T \) |
| 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
−24.7090616161466808503976261842, −24.25512232638237752330775741943, −22.82027308152034567093539381010, −21.71934894365045535910553607119, −21.37062777580798528885167829503, −20.1249827993771741783900139197, −19.67402609870038690197873637101, −18.510677809773057987988370547138, −17.70161377160784577661411635780, −16.75964476684963420700860640681, −15.63922999117464940235311069773, −14.62368150045163266131509848512, −14.01028044502495278828084541885, −13.185853748128165204051284770388, −12.27895964478772839745040512955, −10.83930143482933302874545252071, −9.8806834550204880557621152078, −9.05789682604669672605887051333, −8.35484410376279186902545831257, −6.93394611387093542807073649951, −6.216276440940406941694163724379, −4.71454134922154020569269288269, −3.69264220524575879901192422770, −2.37397941506344628168556101709, −1.59459713622084569339497911385,
1.59459713622084569339497911385, 2.37397941506344628168556101709, 3.69264220524575879901192422770, 4.71454134922154020569269288269, 6.216276440940406941694163724379, 6.93394611387093542807073649951, 8.35484410376279186902545831257, 9.05789682604669672605887051333, 9.8806834550204880557621152078, 10.83930143482933302874545252071, 12.27895964478772839745040512955, 13.185853748128165204051284770388, 14.01028044502495278828084541885, 14.62368150045163266131509848512, 15.63922999117464940235311069773, 16.75964476684963420700860640681, 17.70161377160784577661411635780, 18.510677809773057987988370547138, 19.67402609870038690197873637101, 20.1249827993771741783900139197, 21.37062777580798528885167829503, 21.71934894365045535910553607119, 22.82027308152034567093539381010, 24.25512232638237752330775741943, 24.7090616161466808503976261842