| L(s) = 1 | + 2-s + 3-s + 4-s − 5-s + 6-s + 7-s + 8-s + 9-s − 10-s + 12-s + 14-s − 15-s + 16-s − 17-s + 18-s + 19-s − 20-s + 21-s + 23-s + 24-s + 25-s + 27-s + 28-s − 29-s − 30-s − 31-s + 32-s + ⋯ |
| L(s) = 1 | + 2-s + 3-s + 4-s − 5-s + 6-s + 7-s + 8-s + 9-s − 10-s + 12-s + 14-s − 15-s + 16-s − 17-s + 18-s + 19-s − 20-s + 21-s + 23-s + 24-s + 25-s + 27-s + 28-s − 29-s − 30-s − 31-s + 32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(4.649443604\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.649443604\) |
| \(L(1)\) |
\(\approx\) |
\(2.627131755\) |
| \(L(1)\) |
\(\approx\) |
\(2.627131755\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 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 | \( 1 - T \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−27.96408977759999442643623690551, −26.927457829463296573028306890307, −26.05234989331358239910457461069, −24.540539974553754386750072303886, −24.39636115707601117310346179176, −23.19950491340399864280920917906, −22.07150223966791183044902277884, −20.95388522421390929252004068309, −20.25105000587508602785584474817, −19.47031591354731912249849417448, −18.23402906406616185033460722744, −16.509110580842532185103987657407, −15.37662962692940221671328342424, −14.867548921100765729936823546121, −13.84276467249981041604969414107, −12.81542207779360146951868028515, −11.63116598812765390864877630440, −10.766488725834403000845762565015, −8.94382216950364188850211744182, −7.76050665855674561996549805036, −7.05383785285525996481081470411, −5.08520065823241033479220568574, −4.112131881185385341519118598070, −3.04924425679759808759061989686, −1.61518022087167499869436858277,
1.61518022087167499869436858277, 3.04924425679759808759061989686, 4.112131881185385341519118598070, 5.08520065823241033479220568574, 7.05383785285525996481081470411, 7.76050665855674561996549805036, 8.94382216950364188850211744182, 10.766488725834403000845762565015, 11.63116598812765390864877630440, 12.81542207779360146951868028515, 13.84276467249981041604969414107, 14.867548921100765729936823546121, 15.37662962692940221671328342424, 16.509110580842532185103987657407, 18.23402906406616185033460722744, 19.47031591354731912249849417448, 20.25105000587508602785584474817, 20.95388522421390929252004068309, 22.07150223966791183044902277884, 23.19950491340399864280920917906, 24.39636115707601117310346179176, 24.540539974553754386750072303886, 26.05234989331358239910457461069, 26.927457829463296573028306890307, 27.96408977759999442643623690551
