| L(s) = 1 | + (0.789 + 1.00i)2-s + (−0.304 + 1.70i)3-s + (0.0876 − 0.366i)4-s + (−0.419 + 2.29i)5-s + (−1.94 + 1.04i)6-s + (−2.07 + 1.14i)7-s + (2.74 − 1.27i)8-s + (−2.81 − 1.03i)9-s + (−2.63 + 1.39i)10-s + (−4.36 − 2.84i)11-s + (0.598 + 0.260i)12-s + (−3.29 + 3.00i)13-s + (−2.78 − 1.16i)14-s + (−3.79 − 1.41i)15-s + (2.76 + 1.40i)16-s + (3.51 + 5.83i)17-s + ⋯ |
| L(s) = 1 | + (0.558 + 0.707i)2-s + (−0.175 + 0.984i)3-s + (0.0438 − 0.183i)4-s + (−0.187 + 1.02i)5-s + (−0.794 + 0.425i)6-s + (−0.784 + 0.434i)7-s + (0.971 − 0.449i)8-s + (−0.938 − 0.345i)9-s + (−0.831 + 0.440i)10-s + (−1.31 − 0.857i)11-s + (0.172 + 0.0753i)12-s + (−0.912 + 0.834i)13-s + (−0.745 − 0.312i)14-s + (−0.979 − 0.365i)15-s + (0.692 + 0.351i)16-s + (0.851 + 1.41i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.989 + 0.145i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.989 + 0.145i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0876313 - 1.19924i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0876313 - 1.19924i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.304 - 1.70i)T \) |
| 59 | \( 1 + (-3.99 - 6.56i)T \) |
| good | 2 | \( 1 + (-0.789 - 1.00i)T + (-0.465 + 1.94i)T^{2} \) |
| 5 | \( 1 + (0.419 - 2.29i)T + (-4.67 - 1.76i)T^{2} \) |
| 7 | \( 1 + (2.07 - 1.14i)T + (3.71 - 5.93i)T^{2} \) |
| 11 | \( 1 + (4.36 + 2.84i)T + (4.43 + 10.0i)T^{2} \) |
| 13 | \( 1 + (3.29 - 3.00i)T + (1.17 - 12.9i)T^{2} \) |
| 17 | \( 1 + (-3.51 - 5.83i)T + (-7.96 + 15.0i)T^{2} \) |
| 19 | \( 1 + (0.479 - 1.72i)T + (-16.2 - 9.79i)T^{2} \) |
| 23 | \( 1 + (0.0469 - 0.0967i)T + (-14.2 - 18.0i)T^{2} \) |
| 29 | \( 1 + (-0.687 + 4.72i)T + (-27.7 - 8.26i)T^{2} \) |
| 31 | \( 1 + (-4.44 - 4.36i)T + (0.559 + 30.9i)T^{2} \) |
| 37 | \( 1 + (1.02 - 2.21i)T + (-23.9 - 28.1i)T^{2} \) |
| 41 | \( 1 + (-0.513 + 7.09i)T + (-40.5 - 5.90i)T^{2} \) |
| 43 | \( 1 + (4.54 - 8.96i)T + (-25.4 - 34.6i)T^{2} \) |
| 47 | \( 1 + (1.86 - 0.341i)T + (43.9 - 16.6i)T^{2} \) |
| 53 | \( 1 + (-6.78 - 3.59i)T + (29.7 + 43.8i)T^{2} \) |
| 61 | \( 1 + (-0.281 + 0.222i)T + (14.1 - 59.3i)T^{2} \) |
| 67 | \( 1 + (-3.68 + 5.23i)T + (-22.5 - 63.0i)T^{2} \) |
| 71 | \( 1 + (11.8 - 10.0i)T + (11.4 - 70.0i)T^{2} \) |
| 73 | \( 1 + (-7.31 - 9.61i)T + (-19.5 + 70.3i)T^{2} \) |
| 79 | \( 1 + (-2.54 + 5.76i)T + (-53.2 - 58.3i)T^{2} \) |
| 83 | \( 1 + (-3.16 + 15.7i)T + (-76.5 - 32.1i)T^{2} \) |
| 89 | \( 1 + (0.719 + 1.80i)T + (-64.6 + 61.2i)T^{2} \) |
| 97 | \( 1 + (-4.96 - 11.8i)T + (-67.9 + 69.2i)T^{2} \) |
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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
−11.06214707691457174887411855361, −10.20118008198971431089039048562, −10.02448761374728593922968371721, −8.569759485987095665501849766983, −7.52563085861714931200541114592, −6.36824724781727203319052759228, −5.87758648755878230922430353570, −4.89606282343094077548051956805, −3.70185418348563582705748752734, −2.71564966438998845692832348170,
0.57087027963981957043072093371, 2.33857889237102002453709025363, 3.19055901642828819017854848798, 4.86998210116258893572471707508, 5.24342173222269424724264134169, 6.99770436734995101724892442517, 7.59805323342161324807261681210, 8.338621717610069418778382127831, 9.733970631263180688027032348818, 10.49559333196563804742045355632
