L(s) = 1 | + (0.173 − 0.984i)2-s + (0.173 + 0.984i)3-s + (−0.939 − 0.342i)4-s + (0.766 − 0.642i)5-s + 6-s + (0.766 − 0.642i)7-s + (−0.5 + 0.866i)8-s + (−0.939 + 0.342i)9-s + (−0.5 − 0.866i)10-s + (−0.5 + 0.866i)11-s + (0.173 − 0.984i)12-s + (−0.939 − 0.342i)13-s + (−0.5 − 0.866i)14-s + (0.766 + 0.642i)15-s + (0.766 + 0.642i)16-s + (−0.939 + 0.342i)17-s + ⋯ |
L(s) = 1 | + (0.173 − 0.984i)2-s + (0.173 + 0.984i)3-s + (−0.939 − 0.342i)4-s + (0.766 − 0.642i)5-s + 6-s + (0.766 − 0.642i)7-s + (−0.5 + 0.866i)8-s + (−0.939 + 0.342i)9-s + (−0.5 − 0.866i)10-s + (−0.5 + 0.866i)11-s + (0.173 − 0.984i)12-s + (−0.939 − 0.342i)13-s + (−0.5 − 0.866i)14-s + (0.766 + 0.642i)15-s + (0.766 + 0.642i)16-s + (−0.939 + 0.342i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.721 - 0.691i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.721 - 0.691i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8250224266 - 0.3315152153i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8250224266 - 0.3315152153i\) |
\(L(1)\) |
\(\approx\) |
\(1.025206045 - 0.3186274895i\) |
\(L(1)\) |
\(\approx\) |
\(1.025206045 - 0.3186274895i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 37 | \( 1 \) |
good | 2 | \( 1 + (0.173 - 0.984i)T \) |
| 3 | \( 1 + (0.173 + 0.984i)T \) |
| 5 | \( 1 + (0.766 - 0.642i)T \) |
| 7 | \( 1 + (0.766 - 0.642i)T \) |
| 11 | \( 1 + (-0.5 + 0.866i)T \) |
| 13 | \( 1 + (-0.939 - 0.342i)T \) |
| 17 | \( 1 + (-0.939 + 0.342i)T \) |
| 19 | \( 1 + (0.173 + 0.984i)T \) |
| 23 | \( 1 + (-0.5 - 0.866i)T \) |
| 29 | \( 1 + (-0.5 + 0.866i)T \) |
| 31 | \( 1 + T \) |
| 41 | \( 1 + (-0.939 - 0.342i)T \) |
| 43 | \( 1 + T \) |
| 47 | \( 1 + (-0.5 - 0.866i)T \) |
| 53 | \( 1 + (0.766 + 0.642i)T \) |
| 59 | \( 1 + (0.766 + 0.642i)T \) |
| 61 | \( 1 + (-0.939 - 0.342i)T \) |
| 67 | \( 1 + (0.766 - 0.642i)T \) |
| 71 | \( 1 + (0.173 + 0.984i)T \) |
| 73 | \( 1 + T \) |
| 79 | \( 1 + (0.766 - 0.642i)T \) |
| 83 | \( 1 + (-0.939 + 0.342i)T \) |
| 89 | \( 1 + (0.766 + 0.642i)T \) |
| 97 | \( 1 + (-0.5 - 0.866i)T \) |
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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
−35.55737244664605154613137169247, −34.2809305672477336427861243052, −33.89980765779648276451667289294, −32.09234708713787103886376632073, −31.093774336178057767399289335626, −30.018781440877329512780321028275, −28.684025902822820084645160046464, −26.8296953501195039182700183313, −25.848418219523711037517215212360, −24.59056865419119869131220675196, −24.087985943873179471615206080660, −22.42432591963739819022554320830, −21.38763882005088796796970839023, −19.13023842138606415053999947674, −18.0458315857800586178253031838, −17.33007217366417523049812313020, −15.33804110826129334285532790950, −14.11945426123335035977092320659, −13.30528742196747246570942608615, −11.557289986060654190407696866, −9.23745036491563480214768809695, −7.86234835456625411321924754634, −6.55728926928468506751583600239, −5.31307204237638369371586457279, −2.54193907413823288034334573757,
2.17733868424224259379051329304, 4.3054459372668812777970979105, 5.2340206278592525882109227640, 8.37999505948536789603704018279, 9.833202932226007530465870431, 10.598219017762581822571447456715, 12.33723625771391243172553405852, 13.80586156663537205214950836896, 14.91440701781358440790082633920, 16.92263581474839209067959109801, 17.90049769163092449931166644606, 20.06845691166944875742896842296, 20.59269378898605207965879493732, 21.64437545092678502072254924358, 22.795143236752151450274487423455, 24.39524085695291235329320005029, 26.1691517186216148658972919316, 27.27337132409410490147818940341, 28.297910379631695360519916849738, 29.30528880362276735912024146544, 30.76022308794907941365770727132, 31.857772195123667980447237104573, 32.93643948050253003917803524281, 33.75713042285888893353109045744, 36.071765343100618653714908989790