L(s) = 1 | + (0.939 + 0.342i)3-s + (0.173 − 0.984i)4-s + (0.5 − 0.866i)7-s + (0.766 + 0.642i)9-s + (0.5 − 0.866i)12-s + (−0.939 + 0.342i)13-s + (−0.939 − 0.342i)16-s + (0.766 − 0.642i)21-s + (−0.939 + 0.342i)25-s + (0.500 + 0.866i)27-s + (−0.766 − 0.642i)28-s + (−0.5 + 0.866i)31-s + (0.766 − 0.642i)36-s + 37-s − 39-s + ⋯ |
L(s) = 1 | + (0.939 + 0.342i)3-s + (0.173 − 0.984i)4-s + (0.5 − 0.866i)7-s + (0.766 + 0.642i)9-s + (0.5 − 0.866i)12-s + (−0.939 + 0.342i)13-s + (−0.939 − 0.342i)16-s + (0.766 − 0.642i)21-s + (−0.939 + 0.342i)25-s + (0.500 + 0.866i)27-s + (−0.766 − 0.642i)28-s + (−0.5 + 0.866i)31-s + (0.766 − 0.642i)36-s + 37-s − 39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1083 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.854 + 0.519i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1083 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.854 + 0.519i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.467696454\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.467696454\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.939 - 0.342i)T \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 5 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 7 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (0.939 - 0.342i)T + (0.766 - 0.642i)T^{2} \) |
| 17 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 23 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 29 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 - T + T^{2} \) |
| 41 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 43 | \( 1 + (0.173 + 0.984i)T + (-0.939 + 0.342i)T^{2} \) |
| 47 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 53 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 59 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 61 | \( 1 + (0.173 - 0.984i)T + (-0.939 - 0.342i)T^{2} \) |
| 67 | \( 1 + (-0.766 - 0.642i)T + (0.173 + 0.984i)T^{2} \) |
| 71 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 73 | \( 1 + (-0.939 - 0.342i)T + (0.766 + 0.642i)T^{2} \) |
| 79 | \( 1 + (0.939 + 0.342i)T + (0.766 + 0.642i)T^{2} \) |
| 83 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 97 | \( 1 + (1.53 - 1.28i)T + (0.173 - 0.984i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.906584732226953199096629831618, −9.398543044253590759012514153058, −8.413374431465001918464136749829, −7.43774163392751860193381834109, −6.92543252897288658950375367057, −5.57692685610929721926726518051, −4.69518512219818483484309990436, −3.92560358245955759693853870326, −2.53180443418081366840494250159, −1.49708702052486119361280431217,
2.06098353047006197875299672972, 2.71411987734987024483221287660, 3.78155464067630920508368432177, 4.77985101302363561505232789898, 6.04992242264582748814981224651, 7.09630924138360995887065363023, 7.915670369793766986335233864872, 8.231931380624630159135590671705, 9.241810617774121621737127640632, 9.797606378579731580335932648877