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
−36.071765343100618653714908989790, −33.75713042285888893353109045744, −32.93643948050253003917803524281, −31.857772195123667980447237104573, −30.76022308794907941365770727132, −29.30528880362276735912024146544, −28.297910379631695360519916849738, −27.27337132409410490147818940341, −26.1691517186216148658972919316, −24.39524085695291235329320005029, −22.795143236752151450274487423455, −21.64437545092678502072254924358, −20.59269378898605207965879493732, −20.06845691166944875742896842296, −17.90049769163092449931166644606, −16.92263581474839209067959109801, −14.91440701781358440790082633920, −13.80586156663537205214950836896, −12.33723625771391243172553405852, −10.598219017762581822571447456715, −9.833202932226007530465870431, −8.37999505948536789603704018279, −5.2340206278592525882109227640, −4.3054459372668812777970979105, −2.17733868424224259379051329304,
2.54193907413823288034334573757, 5.31307204237638369371586457279, 6.55728926928468506751583600239, 7.86234835456625411321924754634, 9.23745036491563480214768809695, 11.557289986060654190407696866, 13.30528742196747246570942608615, 14.11945426123335035977092320659, 15.33804110826129334285532790950, 17.33007217366417523049812313020, 18.0458315857800586178253031838, 19.13023842138606415053999947674, 21.38763882005088796796970839023, 22.42432591963739819022554320830, 24.087985943873179471615206080660, 24.59056865419119869131220675196, 25.848418219523711037517215212360, 26.8296953501195039182700183313, 28.684025902822820084645160046464, 30.018781440877329512780321028275, 31.093774336178057767399289335626, 32.09234708713787103886376632073, 33.89980765779648276451667289294, 34.2809305672477336427861243052, 35.55737244664605154613137169247