L(s) = 1 | + (0.879 − 0.475i)3-s + (−0.0825 − 0.996i)5-s + (−0.945 + 0.324i)7-s + (0.546 − 0.837i)9-s + (−0.245 + 0.969i)11-s + (−0.0825 − 0.996i)13-s + (−0.546 − 0.837i)15-s + (−0.0825 − 0.996i)17-s + (0.0825 − 0.996i)19-s + (−0.677 + 0.735i)21-s + (−0.789 − 0.614i)23-s + (−0.986 + 0.164i)25-s + (0.0825 − 0.996i)27-s + (0.945 − 0.324i)29-s + (−0.245 + 0.969i)31-s + ⋯ |
L(s) = 1 | + (0.879 − 0.475i)3-s + (−0.0825 − 0.996i)5-s + (−0.945 + 0.324i)7-s + (0.546 − 0.837i)9-s + (−0.245 + 0.969i)11-s + (−0.0825 − 0.996i)13-s + (−0.546 − 0.837i)15-s + (−0.0825 − 0.996i)17-s + (0.0825 − 0.996i)19-s + (−0.677 + 0.735i)21-s + (−0.789 − 0.614i)23-s + (−0.986 + 0.164i)25-s + (0.0825 − 0.996i)27-s + (0.945 − 0.324i)29-s + (−0.245 + 0.969i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 916 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.636 + 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 916 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.636 + 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.2443923969 - 0.5189637882i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.2443923969 - 0.5189637882i\) |
\(L(1)\) |
\(\approx\) |
\(0.9541521556 - 0.4439122559i\) |
\(L(1)\) |
\(\approx\) |
\(0.9541521556 - 0.4439122559i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 229 | \( 1 \) |
good | 3 | \( 1 + (0.879 - 0.475i)T \) |
| 5 | \( 1 + (-0.0825 - 0.996i)T \) |
| 7 | \( 1 + (-0.945 + 0.324i)T \) |
| 11 | \( 1 + (-0.245 + 0.969i)T \) |
| 13 | \( 1 + (-0.0825 - 0.996i)T \) |
| 17 | \( 1 + (-0.0825 - 0.996i)T \) |
| 19 | \( 1 + (0.0825 - 0.996i)T \) |
| 23 | \( 1 + (-0.789 - 0.614i)T \) |
| 29 | \( 1 + (0.945 - 0.324i)T \) |
| 31 | \( 1 + (-0.245 + 0.969i)T \) |
| 37 | \( 1 + (-0.677 + 0.735i)T \) |
| 41 | \( 1 + (0.546 + 0.837i)T \) |
| 43 | \( 1 + (0.677 - 0.735i)T \) |
| 47 | \( 1 + (-0.546 + 0.837i)T \) |
| 53 | \( 1 + (-0.879 + 0.475i)T \) |
| 59 | \( 1 + (0.677 + 0.735i)T \) |
| 61 | \( 1 + (0.546 - 0.837i)T \) |
| 67 | \( 1 + (-0.546 + 0.837i)T \) |
| 71 | \( 1 + (-0.245 - 0.969i)T \) |
| 73 | \( 1 + (-0.986 + 0.164i)T \) |
| 79 | \( 1 + (-0.945 - 0.324i)T \) |
| 83 | \( 1 + (0.677 + 0.735i)T \) |
| 89 | \( 1 + T \) |
| 97 | \( 1 + (-0.986 - 0.164i)T \) |
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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
−21.96303827154749266587580483150, −21.56713908201214744764815985391, −20.68265643689368971403769612552, −19.50837713810215096524352565175, −19.249611652898388049186512272560, −18.62741030921452985807641803946, −17.4357291496334658731128033930, −16.16592565082317012658037626819, −16.07685177084763438665183122358, −14.891763304219091325013159163120, −14.1938012287008123576653227342, −13.67518150468236627198517270352, −12.73832778938935693396729503478, −11.55534890065789306037704954220, −10.59112297771143742499564587002, −10.073011551209005456294809200898, −9.22610513387054247723977246252, −8.2515548540967894870691242304, −7.4487476369101934249916929305, −6.49322947515697373914878278247, −5.70674625677846763970131746596, −4.024830891264941767223181739723, −3.678225633021611592041381798141, −2.74980141951083367867022709126, −1.75898153528051284690233477145,
0.10766436822159621932611194013, 1.120331796946490626837309152541, 2.42607689750394161685220963501, 3.05540673496342762146731530283, 4.313115293263150879511088385180, 5.14782733131152080063225092996, 6.35528115604925987420767103795, 7.2342601713717352270789841323, 8.069377712746213288013381790512, 8.91218255872063971998020783954, 9.569154895336926812581971922834, 10.29278297656692608417202586941, 11.931991468727213861203095439751, 12.47230679745080797297230958078, 13.08986777876220588833882277154, 13.76606834630499106974642176570, 14.85160710498093912982511392816, 15.836736893464591375958917108167, 15.97507933336792436089403493418, 17.523749892321973258973687470374, 17.93195168041254570389554139115, 19.01861199313971660678896450349, 19.79656370707553761345348274581, 20.2630671813189302418927604037, 20.86035902645936212358690432258