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 + 20-s − 21-s − 22-s + 23-s + 24-s + 25-s + 26-s − 27-s + 28-s − 29-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 + 20-s − 21-s − 22-s + 23-s + 24-s + 25-s + 26-s − 27-s + 28-s − 29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8044195998\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8044195998\) |
\(L(1)\) |
\(\approx\) |
\(0.7207307841\) |
\(L(1)\) |
\(\approx\) |
\(0.7207307841\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 19 | \( 1 \) |
good | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 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
−40.26617802490133472247976643805, −38.82554118183282440132523450561, −37.40813650027561356755811127345, −36.37422478126958236301449933046, −34.84495686055665470774302769745, −33.87521220459731020508729447206, −32.90410771700736502136754493832, −30.126326872819792576033125367508, −29.32412200791730736572299870599, −27.963994956335836722217114002820, −27.03162527243922996956997054359, −25.15657602977425865792582365688, −24.12936469825134045135223793328, −22.01181325277693304291745500774, −20.801408834191412037687964104223, −18.76055592127636504802692087826, −17.415448573735009169608509431565, −16.86361313101393385578914242924, −14.69581220669067395382827430686, −12.176226083345964716923513454085, −10.78581062533769121248302648488, −9.383326327673246749055472537003, −7.16067082490070751048510233990, −5.47661417086705906873501833640, −1.516083753160053883697692705674,
1.516083753160053883697692705674, 5.47661417086705906873501833640, 7.16067082490070751048510233990, 9.383326327673246749055472537003, 10.78581062533769121248302648488, 12.176226083345964716923513454085, 14.69581220669067395382827430686, 16.86361313101393385578914242924, 17.415448573735009169608509431565, 18.76055592127636504802692087826, 20.801408834191412037687964104223, 22.01181325277693304291745500774, 24.12936469825134045135223793328, 25.15657602977425865792582365688, 27.03162527243922996956997054359, 27.963994956335836722217114002820, 29.32412200791730736572299870599, 30.126326872819792576033125367508, 32.90410771700736502136754493832, 33.87521220459731020508729447206, 34.84495686055665470774302769745, 36.37422478126958236301449933046, 37.40813650027561356755811127345, 38.82554118183282440132523450561, 40.26617802490133472247976643805