L(s) = 1 | + (1.22 + 0.707i)2-s + (−0.866 − 1.5i)3-s + (0.999 + 1.73i)4-s + (1.95 + 4.60i)5-s − 2.44i·6-s + (−2.31 + 6.60i)7-s + 2.82i·8-s + (−1.5 + 2.59i)9-s + (−0.866 + 7.01i)10-s + (−0.676 − 1.17i)11-s + (1.73 − 2.99i)12-s − 5.90·13-s + (−7.50 + 6.45i)14-s + (5.21 − 6.91i)15-s + (−2.00 + 3.46i)16-s + (6.21 + 10.7i)17-s + ⋯ |
L(s) = 1 | + (0.612 + 0.353i)2-s + (−0.288 − 0.5i)3-s + (0.249 + 0.433i)4-s + (0.390 + 0.920i)5-s − 0.408i·6-s + (−0.330 + 0.943i)7-s + 0.353i·8-s + (−0.166 + 0.288i)9-s + (−0.0866 + 0.701i)10-s + (−0.0615 − 0.106i)11-s + (0.144 − 0.249i)12-s − 0.454·13-s + (−0.536 + 0.461i)14-s + (0.347 − 0.460i)15-s + (−0.125 + 0.216i)16-s + (0.365 + 0.633i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0629 - 0.998i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0629 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.38211 + 1.29763i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.38211 + 1.29763i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.22 - 0.707i)T \) |
| 3 | \( 1 + (0.866 + 1.5i)T \) |
| 5 | \( 1 + (-1.95 - 4.60i)T \) |
| 7 | \( 1 + (2.31 - 6.60i)T \) |
good | 11 | \( 1 + (0.676 + 1.17i)T + (-60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + 5.90T + 169T^{2} \) |
| 17 | \( 1 + (-6.21 - 10.7i)T + (-144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (-27.5 - 15.9i)T + (180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (1.31 + 0.758i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + 15.0T + 841T^{2} \) |
| 31 | \( 1 + (-40.3 + 23.2i)T + (480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + (0.441 + 0.254i)T + (684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + 36.0iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 67.6iT - 1.84e3T^{2} \) |
| 47 | \( 1 + (-24.7 + 42.9i)T + (-1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + (45.5 - 26.3i)T + (1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + (62.0 - 35.8i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-50.6 - 29.2i)T + (1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-109. + 63.1i)T + (2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 + 55.4T + 5.04e3T^{2} \) |
| 73 | \( 1 + (-26.5 - 46.0i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-19.9 + 34.5i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 - 120.T + 6.88e3T^{2} \) |
| 89 | \( 1 + (5.68 + 3.28i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 - 154.T + 9.40e3T^{2} \) |
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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
−12.26770828627651703280844027949, −11.79317521979337552894024239138, −10.53765399630843215673258385724, −9.497892823610159001761572921151, −8.057235888813629631505375383387, −7.09140463557851314014715274277, −6.03736480021409702457615873875, −5.41325854982743154469417504373, −3.46832512222599188348360791601, −2.23431878244982105308773613466,
0.956197477784741632634172423895, 3.08988344651255609204872793216, 4.53187755459247296311813554406, 5.19275279270203564442108563250, 6.52400794961985451880711709898, 7.79856629750504757695160874691, 9.483572895522158511293117138292, 9.823739390037969596587665467939, 11.05930433517990723741849867770, 11.94885644227978360607515277363