L(s) = 1 | + (−0.766 + 0.642i)2-s + (−0.766 + 0.642i)3-s + (0.173 − 0.984i)4-s + (0.173 + 0.984i)5-s + (0.173 − 0.984i)6-s + (0.5 + 0.866i)8-s + (0.173 − 0.984i)9-s + (−0.766 − 0.642i)10-s + (−0.5 + 0.866i)11-s + (0.5 + 0.866i)12-s + (−0.173 + 0.984i)13-s + (−0.766 − 0.642i)15-s + (−0.939 − 0.342i)16-s + (0.173 + 0.984i)17-s + (0.5 + 0.866i)18-s + ⋯ |
L(s) = 1 | + (−0.766 + 0.642i)2-s + (−0.766 + 0.642i)3-s + (0.173 − 0.984i)4-s + (0.173 + 0.984i)5-s + (0.173 − 0.984i)6-s + (0.5 + 0.866i)8-s + (0.173 − 0.984i)9-s + (−0.766 − 0.642i)10-s + (−0.5 + 0.866i)11-s + (0.5 + 0.866i)12-s + (−0.173 + 0.984i)13-s + (−0.766 − 0.642i)15-s + (−0.939 − 0.342i)16-s + (0.173 + 0.984i)17-s + (0.5 + 0.866i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.258 - 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.258 - 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.1986578534 + 0.2587020270i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.1986578534 + 0.2587020270i\) |
\(L(1)\) |
\(\approx\) |
\(0.3547882779 + 0.3553022951i\) |
\(L(1)\) |
\(\approx\) |
\(0.3547882779 + 0.3553022951i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + (-0.766 + 0.642i)T \) |
| 3 | \( 1 + (-0.766 + 0.642i)T \) |
| 5 | \( 1 + (0.173 + 0.984i)T \) |
| 11 | \( 1 + (-0.5 + 0.866i)T \) |
| 13 | \( 1 + (-0.173 + 0.984i)T \) |
| 17 | \( 1 + (0.173 + 0.984i)T \) |
| 23 | \( 1 + (-0.939 + 0.342i)T \) |
| 29 | \( 1 + (0.939 - 0.342i)T \) |
| 31 | \( 1 - T \) |
| 37 | \( 1 + (0.5 - 0.866i)T \) |
| 41 | \( 1 + (-0.173 - 0.984i)T \) |
| 43 | \( 1 + (0.766 - 0.642i)T \) |
| 47 | \( 1 + (0.173 - 0.984i)T \) |
| 53 | \( 1 + (-0.173 + 0.984i)T \) |
| 59 | \( 1 + (-0.173 - 0.984i)T \) |
| 61 | \( 1 + (-0.939 + 0.342i)T \) |
| 67 | \( 1 + (-0.766 - 0.642i)T \) |
| 71 | \( 1 + (-0.766 + 0.642i)T \) |
| 73 | \( 1 + (0.766 - 0.642i)T \) |
| 79 | \( 1 + (0.939 + 0.342i)T \) |
| 83 | \( 1 + (-0.5 + 0.866i)T \) |
| 89 | \( 1 + (-0.766 - 0.642i)T \) |
| 97 | \( 1 + (0.939 + 0.342i)T \) |
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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
−27.74812227966763575602320028688, −27.15798365972448041583659297729, −25.57628400403188800574387917932, −24.74989833860488912822839699263, −23.83519080765521179002726541675, −22.504247204596445576875268961850, −21.52583722863078291908501388336, −20.43064490070523201311944503946, −19.54563165901299756176749742418, −18.32788116181994504411561623932, −17.72640945167970335508993515627, −16.542168176866921846602528567045, −16.04019497024209063020770619751, −13.654461620761983083570861577347, −12.79994598560872433667108218538, −11.97589572145809585339903946158, −10.9119244672462201187726200439, −9.80980630483832482601132102398, −8.41057540578297726733309957174, −7.62267497398128350632701815399, −6.004026348950445649080495999258, −4.74862730674084547418092248278, −2.78812362470707168631759427869, −1.209021009728704343135912637979, −0.201987131893221278106500233852,
2.00346901352485149413154085320, 4.128220417018680593443079801449, 5.58280505070058914720979040360, 6.54986323620669505308992072907, 7.55457642526334992058606083868, 9.23296539276153449337494685824, 10.18717832928875051023445018532, 10.86198787217270702161828177287, 12.117453262412407599239044719582, 14.06982341380104597577863798878, 15.03080733166170500270304672019, 15.80955915126873575712101506958, 16.96352304239156553388086757166, 17.79366582249340502010297055520, 18.58183642382065312048742905781, 19.777500750158288881404235744878, 21.19500578773903892313458301011, 22.1903627238195044854740993253, 23.29133528982652101413914480248, 23.87715701386129625031846917084, 25.519125264474671351581595815131, 26.216997223340428140399877205196, 26.94106775152865568219735998656, 28.03378684954286657112962742845, 28.74504183514368514318565214647