| L(s) = 1 | + (−0.878 − 2.41i)2-s + (1.25 + 1.19i)3-s + (−3.52 + 2.95i)4-s + (0.588 + 3.33i)5-s + (1.78 − 4.07i)6-s + (−2.64 − 0.108i)7-s + (5.79 + 3.34i)8-s + (0.145 + 2.99i)9-s + (7.53 − 4.35i)10-s + (4.55 + 0.803i)11-s + (−7.95 − 0.502i)12-s + (0.619 − 1.70i)13-s + (2.06 + 6.47i)14-s + (−3.24 + 4.88i)15-s + (1.38 − 7.86i)16-s + (1.40 + 2.43i)17-s + ⋯ |
| L(s) = 1 | + (−0.621 − 1.70i)2-s + (0.724 + 0.689i)3-s + (−1.76 + 1.47i)4-s + (0.263 + 1.49i)5-s + (0.727 − 1.66i)6-s + (−0.999 − 0.0411i)7-s + (2.04 + 1.18i)8-s + (0.0484 + 0.998i)9-s + (2.38 − 1.37i)10-s + (1.37 + 0.242i)11-s + (−2.29 − 0.145i)12-s + (0.171 − 0.472i)13-s + (0.550 + 1.73i)14-s + (−0.838 + 1.26i)15-s + (0.346 − 1.96i)16-s + (0.340 + 0.590i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.113i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 + 0.113i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.951173 - 0.0542621i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.951173 - 0.0542621i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.25 - 1.19i)T \) |
| 7 | \( 1 + (2.64 + 0.108i)T \) |
| good | 2 | \( 1 + (0.878 + 2.41i)T + (-1.53 + 1.28i)T^{2} \) |
| 5 | \( 1 + (-0.588 - 3.33i)T + (-4.69 + 1.71i)T^{2} \) |
| 11 | \( 1 + (-4.55 - 0.803i)T + (10.3 + 3.76i)T^{2} \) |
| 13 | \( 1 + (-0.619 + 1.70i)T + (-9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (-1.40 - 2.43i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.586 + 0.338i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (2.29 + 2.74i)T + (-3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (0.546 + 1.50i)T + (-22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (1.99 + 2.37i)T + (-5.38 + 30.5i)T^{2} \) |
| 37 | \( 1 + (0.898 + 1.55i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-11.6 - 4.22i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (0.0534 - 0.303i)T + (-40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (2.22 + 1.86i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + 4.31iT - 53T^{2} \) |
| 59 | \( 1 + (1.34 + 7.61i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (-0.783 + 0.933i)T + (-10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (-0.826 - 0.300i)T + (51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (-11.6 + 6.72i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (11.2 + 6.52i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-13.7 + 5.01i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (-6.55 + 2.38i)T + (63.5 - 53.3i)T^{2} \) |
| 89 | \( 1 + (8.53 - 14.7i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-5.81 - 1.02i)T + (91.1 + 33.1i)T^{2} \) |
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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
−12.35324014026927901949832990012, −11.15144021208638057583477480504, −10.50620757735568102928587017838, −9.757745095267167403506530215533, −9.187858247133961807754134741820, −7.87634511834049994882166173885, −6.39230842243537410737253712388, −3.98111981057825688709764747309, −3.31836774839629647558021969155, −2.25507898501513281775033542344,
1.08842236545674502273874572896, 4.06609034250180841333060917873, 5.64887890655598412213104375810, 6.51033406074116657990973566505, 7.46237800175939707117171386501, 8.631108476441937895247294309927, 9.170593030039226069549583030749, 9.643136184995127570231201346329, 12.03544229974337884738507000221, 12.93406771300669760561398993717
