L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s + 7-s − 8-s + 9-s + 10-s + 11-s + 12-s − 13-s − 14-s − 15-s + 16-s − 17-s − 18-s + 19-s − 20-s + 21-s − 22-s + 23-s − 24-s + 25-s + 26-s + 27-s + 28-s + ⋯ |
L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s + 7-s − 8-s + 9-s + 10-s + 11-s + 12-s − 13-s − 14-s − 15-s + 16-s − 17-s − 18-s + 19-s − 20-s + 21-s − 22-s + 23-s − 24-s + 25-s + 26-s + 27-s + 28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 227 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 227 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.746062087\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.746062087\) |
\(L(1)\) |
\(\approx\) |
\(1.042574139\) |
\(L(1)\) |
\(\approx\) |
\(1.042574139\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 227 | \( 1 \) |
good | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 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
−26.512061577034567642790675601, −25.232294821693138918001093951, −24.461901395436914834422505747211, −23.99934175009914856513859678556, −22.275693809268395778558549871019, −21.13472827219716190192739824318, −20.10052102541872956701878073610, −19.74417749994118174083404945328, −18.81857565034216215600169537340, −17.81831877851907389331928539727, −16.79069219842832826952037451896, −15.5832005101008724828688303754, −14.9620935623522753043033965611, −14.08366466071811479349226165518, −12.357722978746647108889391658857, −11.570946556842816551920995940405, −10.51975656754414363669118992126, −9.16143782949255695075342973275, −8.60355748416399363056897584061, −7.49087200252593276936746632080, −6.98485679544884034211125028637, −4.81687184042712103618980088058, −3.5390428048487967663480720486, −2.26146006995272507036991556192, −0.98380091960824979110424396224,
0.98380091960824979110424396224, 2.26146006995272507036991556192, 3.5390428048487967663480720486, 4.81687184042712103618980088058, 6.98485679544884034211125028637, 7.49087200252593276936746632080, 8.60355748416399363056897584061, 9.16143782949255695075342973275, 10.51975656754414363669118992126, 11.570946556842816551920995940405, 12.357722978746647108889391658857, 14.08366466071811479349226165518, 14.9620935623522753043033965611, 15.5832005101008724828688303754, 16.79069219842832826952037451896, 17.81831877851907389331928539727, 18.81857565034216215600169537340, 19.74417749994118174083404945328, 20.10052102541872956701878073610, 21.13472827219716190192739824318, 22.275693809268395778558549871019, 23.99934175009914856513859678556, 24.461901395436914834422505747211, 25.232294821693138918001093951, 26.512061577034567642790675601