L(s) = 1 | + (−0.989 + 0.145i)2-s + (0.694 − 0.719i)3-s + (0.957 − 0.288i)4-s + (−0.833 − 0.551i)5-s + (−0.581 + 0.813i)6-s + (−0.252 + 0.967i)7-s + (−0.905 + 0.424i)8-s + (−0.0365 − 0.999i)9-s + (0.905 + 0.424i)10-s + (0.976 − 0.217i)11-s + (0.457 − 0.889i)12-s + (0.989 − 0.145i)13-s + (0.109 − 0.994i)14-s + (−0.976 + 0.217i)15-s + (0.833 − 0.551i)16-s + (0.322 − 0.946i)17-s + ⋯ |
L(s) = 1 | + (−0.989 + 0.145i)2-s + (0.694 − 0.719i)3-s + (0.957 − 0.288i)4-s + (−0.833 − 0.551i)5-s + (−0.581 + 0.813i)6-s + (−0.252 + 0.967i)7-s + (−0.905 + 0.424i)8-s + (−0.0365 − 0.999i)9-s + (0.905 + 0.424i)10-s + (0.976 − 0.217i)11-s + (0.457 − 0.889i)12-s + (0.989 − 0.145i)13-s + (0.109 − 0.994i)14-s + (−0.976 + 0.217i)15-s + (0.833 − 0.551i)16-s + (0.322 − 0.946i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.214 - 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.214 - 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6184085008 - 0.4971548785i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6184085008 - 0.4971548785i\) |
\(L(1)\) |
\(\approx\) |
\(0.7410687735 - 0.2607641046i\) |
\(L(1)\) |
\(\approx\) |
\(0.7410687735 - 0.2607641046i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 173 | \( 1 \) |
good | 2 | \( 1 + (-0.989 + 0.145i)T \) |
| 3 | \( 1 + (0.694 - 0.719i)T \) |
| 5 | \( 1 + (-0.833 - 0.551i)T \) |
| 7 | \( 1 + (-0.252 + 0.967i)T \) |
| 11 | \( 1 + (0.976 - 0.217i)T \) |
| 13 | \( 1 + (0.989 - 0.145i)T \) |
| 17 | \( 1 + (0.322 - 0.946i)T \) |
| 19 | \( 1 + (-0.109 - 0.994i)T \) |
| 23 | \( 1 + (-0.976 - 0.217i)T \) |
| 29 | \( 1 + (-0.581 - 0.813i)T \) |
| 31 | \( 1 + (-0.694 - 0.719i)T \) |
| 37 | \( 1 + (0.109 + 0.994i)T \) |
| 41 | \( 1 + (0.252 - 0.967i)T \) |
| 43 | \( 1 + (0.957 + 0.288i)T \) |
| 47 | \( 1 + (-0.997 - 0.0729i)T \) |
| 53 | \( 1 + (0.791 - 0.611i)T \) |
| 59 | \( 1 + (0.872 + 0.489i)T \) |
| 61 | \( 1 + (0.322 + 0.946i)T \) |
| 67 | \( 1 + (-0.694 + 0.719i)T \) |
| 71 | \( 1 + (0.581 + 0.813i)T \) |
| 73 | \( 1 + (0.639 + 0.768i)T \) |
| 79 | \( 1 + (0.997 - 0.0729i)T \) |
| 83 | \( 1 + (0.391 - 0.920i)T \) |
| 89 | \( 1 + (0.744 + 0.667i)T \) |
| 97 | \( 1 + (-0.391 - 0.920i)T \) |
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
−27.56276883584828244253210775472, −26.74899317327818161215423834876, −26.01710819732143840138916194968, −25.35963550693322656199347445088, −23.95017014296941126846975476713, −22.833829430239479016780697407066, −21.657655490954000312485516905565, −20.551309775055080334536398537383, −19.780299259220716127665078660216, −19.24907419035398520620298162449, −18.0500270693028219799935207571, −16.607192527023211148839324259495, −16.17405933508675260130726053665, −14.94455065742587242600165201661, −14.168435173549221604036149490986, −12.487208622581610933522411008064, −11.09763288068638687238022230981, −10.51017049169835028672245094838, −9.45160927545054649018823597169, −8.30818092409795279945331226011, −7.507922225867234381976028845183, −6.31242895870243611509879614554, −3.82482801190383537978987060871, −3.589650608077073913132161133947, −1.66196961330314016477414825564,
0.885158022172337517242732315072, 2.35554801054246730978071218684, 3.67949387565887868106700940667, 5.81502299586467511746437435373, 6.91431389670751079248601042464, 8.04104278963309820292263014384, 8.823265174517199560772652363210, 9.47491161818611898878272608977, 11.46770743418195172502839736924, 11.94863276359665904263636668385, 13.211394938870478842676341447964, 14.67412921232382197530187750675, 15.57716466181677795854174936886, 16.391792651428133266906122553438, 17.74908151096491099065088020067, 18.6749988458240108981209134698, 19.32298203217730205599674953662, 20.16946519809400533092970543275, 20.96862270607582432119416111606, 22.61889556526429132013838548955, 24.02428014551065880307981390109, 24.49153353391565041923135503630, 25.44740751153277826869682638791, 26.133774099547005973764027159537, 27.45534740167061387394986987553