| L(s) = 1 | + (2.16 − 0.788i)2-s + (−0.782 + 1.54i)3-s + (2.53 − 2.13i)4-s + (0.276 + 1.56i)5-s + (−0.476 + 3.96i)6-s + (0.766 + 0.642i)7-s + (1.51 − 2.62i)8-s + (−1.77 − 2.41i)9-s + (1.83 + 3.18i)10-s + (0.665 − 3.77i)11-s + (1.30 + 5.59i)12-s + (−2.53 − 0.923i)13-s + (2.16 + 0.788i)14-s + (−2.64 − 0.799i)15-s + (0.0619 − 0.351i)16-s + (−1.47 − 2.55i)17-s + ⋯ |
| L(s) = 1 | + (1.53 − 0.557i)2-s + (−0.451 + 0.892i)3-s + (1.26 − 1.06i)4-s + (0.123 + 0.701i)5-s + (−0.194 + 1.61i)6-s + (0.289 + 0.242i)7-s + (0.535 − 0.927i)8-s + (−0.591 − 0.805i)9-s + (0.580 + 1.00i)10-s + (0.200 − 1.13i)11-s + (0.376 + 1.61i)12-s + (−0.704 − 0.256i)13-s + (0.578 + 0.210i)14-s + (−0.681 − 0.206i)15-s + (0.0154 − 0.0877i)16-s + (−0.358 − 0.620i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.106i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 - 0.106i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.21301 + 0.118103i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.21301 + 0.118103i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.782 - 1.54i)T \) |
| 7 | \( 1 + (-0.766 - 0.642i)T \) |
| good | 2 | \( 1 + (-2.16 + 0.788i)T + (1.53 - 1.28i)T^{2} \) |
| 5 | \( 1 + (-0.276 - 1.56i)T + (-4.69 + 1.71i)T^{2} \) |
| 11 | \( 1 + (-0.665 + 3.77i)T + (-10.3 - 3.76i)T^{2} \) |
| 13 | \( 1 + (2.53 + 0.923i)T + (9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (1.47 + 2.55i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.47 + 2.55i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (2.49 - 2.09i)T + (3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (6.77 - 2.46i)T + (22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (3.47 - 2.91i)T + (5.38 - 30.5i)T^{2} \) |
| 37 | \( 1 + (-4.88 - 8.45i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (4.91 + 1.78i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (-2.16 + 12.2i)T + (-40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (-4.32 - 3.62i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + 4.32T + 53T^{2} \) |
| 59 | \( 1 + (1.02 + 5.80i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (-8.74 - 7.34i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (-11.0 - 4.00i)T + (51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (2.86 + 4.96i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (6.41 - 11.1i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-4.96 + 1.80i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (-1.95 + 0.711i)T + (63.5 - 53.3i)T^{2} \) |
| 89 | \( 1 + (6.51 - 11.2i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-2.45 + 13.8i)T + (-91.1 - 33.1i)T^{2} \) |
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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
−12.47521775388384451574153355596, −11.45235982474354760936085144847, −11.11977829994755008909179672183, −10.08550991630064463633388379478, −8.811444889279211992358559086003, −6.89662372292771586455006617474, −5.72794833722606576403674689784, −5.01822183932216083016760691583, −3.70435148371053698620759754135, −2.74065333156552390531680050438,
2.06656469255931437348488261322, 4.16813122235398212176542198922, 5.09346174041008225977035640679, 6.06859356115254144080134294192, 7.13658059529451475613731734832, 7.898981771547923468003859568860, 9.542512712909587847717123269264, 11.14712373425074934090861764125, 12.14276393248986378818477026274, 12.71341523212649290090158919308
