L(s) = 1 | + (−0.618 + 0.785i)2-s + (0.869 − 0.493i)3-s + (−0.234 − 0.972i)4-s + (−0.744 − 0.668i)5-s + (−0.150 + 0.988i)6-s + (0.651 − 0.758i)7-s + (0.908 + 0.417i)8-s + (0.512 − 0.858i)9-s + (0.985 − 0.171i)10-s + (0.966 − 0.255i)11-s + (−0.683 − 0.729i)12-s + (−0.954 − 0.296i)13-s + (0.192 + 0.981i)14-s + (−0.976 − 0.213i)15-s + (−0.890 + 0.455i)16-s + (−0.618 + 0.785i)17-s + ⋯ |
L(s) = 1 | + (−0.618 + 0.785i)2-s + (0.869 − 0.493i)3-s + (−0.234 − 0.972i)4-s + (−0.744 − 0.668i)5-s + (−0.150 + 0.988i)6-s + (0.651 − 0.758i)7-s + (0.908 + 0.417i)8-s + (0.512 − 0.858i)9-s + (0.985 − 0.171i)10-s + (0.966 − 0.255i)11-s + (−0.683 − 0.729i)12-s + (−0.954 − 0.296i)13-s + (0.192 + 0.981i)14-s + (−0.976 − 0.213i)15-s + (−0.890 + 0.455i)16-s + (−0.618 + 0.785i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 293 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.316 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 293 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.316 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8246757493 - 0.5943338431i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8246757493 - 0.5943338431i\) |
\(L(1)\) |
\(\approx\) |
\(0.9124327063 - 0.1841539498i\) |
\(L(1)\) |
\(\approx\) |
\(0.9124327063 - 0.1841539498i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 293 | \( 1 \) |
good | 2 | \( 1 + (-0.618 + 0.785i)T \) |
| 3 | \( 1 + (0.869 - 0.493i)T \) |
| 5 | \( 1 + (-0.744 - 0.668i)T \) |
| 7 | \( 1 + (0.651 - 0.758i)T \) |
| 11 | \( 1 + (0.966 - 0.255i)T \) |
| 13 | \( 1 + (-0.954 - 0.296i)T \) |
| 17 | \( 1 + (-0.618 + 0.785i)T \) |
| 19 | \( 1 + (-0.976 - 0.213i)T \) |
| 23 | \( 1 + (-0.150 - 0.988i)T \) |
| 29 | \( 1 + (0.941 - 0.337i)T \) |
| 31 | \( 1 + (-0.744 + 0.668i)T \) |
| 37 | \( 1 + (0.584 - 0.811i)T \) |
| 41 | \( 1 + (-0.847 + 0.530i)T \) |
| 43 | \( 1 + (0.276 + 0.961i)T \) |
| 47 | \( 1 + (-0.397 - 0.917i)T \) |
| 53 | \( 1 + (0.966 - 0.255i)T \) |
| 59 | \( 1 + (-0.0645 - 0.997i)T \) |
| 61 | \( 1 + (0.996 + 0.0859i)T \) |
| 67 | \( 1 + (-0.991 - 0.128i)T \) |
| 71 | \( 1 + (0.714 + 0.699i)T \) |
| 73 | \( 1 + (-0.234 + 0.972i)T \) |
| 79 | \( 1 + (0.996 + 0.0859i)T \) |
| 83 | \( 1 + (0.0215 - 0.999i)T \) |
| 89 | \( 1 + (0.996 - 0.0859i)T \) |
| 97 | \( 1 + (-0.925 + 0.377i)T \) |
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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
−25.709088625856300747125803584804, −25.16277989729376252110486333575, −23.95827318370869309611177401178, −22.317703580997621704205102156665, −22.0002620715660212644041029560, −21.01412446282252181922172378882, −19.99499789914999681576013276288, −19.423542306948096007693967423925, −18.677547848599395763088972681364, −17.67601787951235901387380214874, −16.54223775843375985232646829798, −15.371631846910320664995313973578, −14.70294584947026957068276323847, −13.7071068688659082897860040074, −12.231107630913334251599447796917, −11.58464416570279614103687631607, −10.605860394160063501861091989482, −9.51313176451555867515506794208, −8.781062596188825222951237338589, −7.82941503715973865373563063067, −6.94196531359161726704290818786, −4.72602344912232845511656375214, −3.90856220614720318598590935091, −2.73189575635126415722855815808, −1.912551505773731648161637567028,
0.7664195713823130374570593117, 1.973542200368101956886645122057, 3.95908761192970370910163270101, 4.73702946990192561343768516827, 6.48414473408317927282610307662, 7.283686415371794071138863516733, 8.31796081745970804389403681914, 8.66391384270837328052136718101, 9.919756069787236666537760685563, 11.115325569734835324262742304716, 12.41505415311037780052077723901, 13.42449451027337835724166856262, 14.66006018468707028784356736617, 14.8521783831384530242967863900, 16.21835041672187804726960145171, 17.11122153385784474859932529352, 17.81951128262457853607705147314, 19.122565576463607501091276399806, 19.80615212672818089557911766147, 20.14446187189221780348828931196, 21.560727177193300070407481432884, 23.11489921393115264532412082260, 23.833149535340372425453342016368, 24.55114646943907279851301027034, 25.0102305218622810052376559661