L(s) = 1 | + (−0.939 − 0.342i)2-s + (0.766 + 0.642i)4-s + (−0.939 + 0.342i)5-s + (−0.5 − 0.866i)8-s + 10-s + (−0.939 − 0.342i)11-s + (0.173 − 0.984i)13-s + (0.173 + 0.984i)16-s + 17-s + 19-s + (−0.939 − 0.342i)20-s + (0.766 + 0.642i)22-s + (0.173 − 0.984i)23-s + (0.766 − 0.642i)25-s + (−0.5 + 0.866i)26-s + ⋯ |
L(s) = 1 | + (−0.939 − 0.342i)2-s + (0.766 + 0.642i)4-s + (−0.939 + 0.342i)5-s + (−0.5 − 0.866i)8-s + 10-s + (−0.939 − 0.342i)11-s + (0.173 − 0.984i)13-s + (0.173 + 0.984i)16-s + 17-s + 19-s + (−0.939 − 0.342i)20-s + (0.766 + 0.642i)22-s + (0.173 − 0.984i)23-s + (0.766 − 0.642i)25-s + (−0.5 + 0.866i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.545 - 0.838i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.545 - 0.838i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5144427261 - 0.2791130855i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5144427261 - 0.2791130855i\) |
\(L(1)\) |
\(\approx\) |
\(0.6006522611 - 0.1314355681i\) |
\(L(1)\) |
\(\approx\) |
\(0.6006522611 - 0.1314355681i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (-0.939 - 0.342i)T \) |
| 5 | \( 1 + (-0.939 + 0.342i)T \) |
| 11 | \( 1 + (-0.939 - 0.342i)T \) |
| 13 | \( 1 + (0.173 - 0.984i)T \) |
| 17 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| 23 | \( 1 + (0.173 - 0.984i)T \) |
| 29 | \( 1 + (0.173 + 0.984i)T \) |
| 31 | \( 1 + (0.766 + 0.642i)T \) |
| 37 | \( 1 + (-0.5 - 0.866i)T \) |
| 41 | \( 1 + (0.173 - 0.984i)T \) |
| 43 | \( 1 + (0.766 - 0.642i)T \) |
| 47 | \( 1 + (0.766 - 0.642i)T \) |
| 53 | \( 1 + (-0.5 - 0.866i)T \) |
| 59 | \( 1 + (0.173 - 0.984i)T \) |
| 61 | \( 1 + (0.766 - 0.642i)T \) |
| 67 | \( 1 + (-0.939 + 0.342i)T \) |
| 71 | \( 1 + (-0.5 + 0.866i)T \) |
| 73 | \( 1 + (-0.5 + 0.866i)T \) |
| 79 | \( 1 + (-0.939 - 0.342i)T \) |
| 83 | \( 1 + (0.173 + 0.984i)T \) |
| 89 | \( 1 + T \) |
| 97 | \( 1 + (0.766 - 0.642i)T \) |
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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
−27.18628859654029698990759131280, −26.4296265519772798177573868871, −25.58261864158159988431816720816, −24.40995966823279463209489645732, −23.686147280655010037986167862427, −22.919201958308423557005558418124, −21.12959458965818261799097697836, −20.46779214404963950743530696384, −19.28867932493650579462633087577, −18.758698122643996233614294941933, −17.60160756031900609992598107667, −16.50151279725860415837727062003, −15.81232434128355591041715657759, −14.97135655049408693991920043235, −13.63683142537748918555345857386, −12.06359753865536318250348636787, −11.40321663523085890895677475317, −10.10723206021266267385152531567, −9.15182008982845331635502133744, −7.895046344501581337369057722775, −7.405860239568665956514192510895, −5.90045455740010423381968663317, −4.61942918058541943115682726740, −2.94941246658944297121807866674, −1.22940375067293220935116064291,
0.74056409834025773810068304142, 2.73935755543480026201934584324, 3.56795433399936253462246625545, 5.41385503361266670946259417620, 7.027774811150753794459927024656, 7.87034711158664194138820342669, 8.67800195295663848465760867340, 10.23701105511327083215519058695, 10.79143174387981890378166793283, 11.98645263258063077126640292865, 12.77493132791118581961315758262, 14.40410977217697404844705775152, 15.75009679165892306597357496142, 16.10489705465507071220661799017, 17.51934018377438934936481500146, 18.482171446082792808421340029879, 19.09419278499188237650785708749, 20.219230586892338859677766084682, 20.86047623792057412722306789191, 22.15902554489442539153800162447, 23.17541043733494271196549476884, 24.252394311043475123911617080668, 25.30690771258804974359022491641, 26.32326454925432256351871370589, 26.9885264597529657019960106472