L(s) = 1 | + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + (0.5 + 0.866i)5-s − 7-s + 0.999·8-s + (0.499 − 0.866i)10-s + 4.77·11-s + (−1.88 + 3.26i)13-s + (0.5 + 0.866i)14-s + (−0.5 − 0.866i)16-s + (−1.38 + 4.13i)19-s − 0.999·20-s + (−2.38 − 4.13i)22-s + (2.38 − 4.13i)23-s + (−0.499 + 0.866i)25-s + 3.77·26-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.223 + 0.387i)5-s − 0.377·7-s + 0.353·8-s + (0.158 − 0.273i)10-s + 1.43·11-s + (−0.523 + 0.906i)13-s + (0.133 + 0.231i)14-s + (−0.125 − 0.216i)16-s + (−0.317 + 0.948i)19-s − 0.223·20-s + (−0.508 − 0.881i)22-s + (0.497 − 0.861i)23-s + (−0.0999 + 0.173i)25-s + 0.739·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.511 - 0.859i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.511 - 0.859i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.131718287\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.131718287\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 + 0.866i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-0.5 - 0.866i)T \) |
| 19 | \( 1 + (1.38 - 4.13i)T \) |
good | 7 | \( 1 + T + 7T^{2} \) |
| 11 | \( 1 - 4.77T + 11T^{2} \) |
| 13 | \( 1 + (1.88 - 3.26i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-8.5 + 14.7i)T^{2} \) |
| 23 | \( 1 + (-2.38 + 4.13i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (3 - 5.19i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 5.77T + 31T^{2} \) |
| 37 | \( 1 + T + 37T^{2} \) |
| 41 | \( 1 + (-2.38 - 4.13i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (0.113 + 0.197i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.613 + 1.06i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.77 - 3.06i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.88 + 4.99i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.11 + 1.92i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-4.77 - 8.26i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-5.88 - 10.1i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-7.65 - 13.2i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 6T + 83T^{2} \) |
| 89 | \( 1 + (4.15 - 7.20i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-5 - 8.66i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.440773083837793887133102147220, −8.981964311888335274907445429545, −8.026955140118185158819634174532, −6.87173065501812572966564888380, −6.58743054262481575492116459841, −5.34919018200866824425046521707, −4.16373742526500491047557226056, −3.53745982503911188745736936020, −2.33982749232990276966922484867, −1.36077992256679953459763631584,
0.50884624808804655187457404958, 1.85472398815646881028486209826, 3.29603934635921034149688120744, 4.32464486195986840112859820405, 5.29357734397017709618117949084, 6.03056003461046428143383736529, 6.87513848425721295766566450959, 7.53717956067928069934400461723, 8.479953025613688019908380402031, 9.374971130811312343471831773466