| L(s) = 1 | + (1.22 − 0.707i)2-s + (−0.866 + 1.5i)3-s + (0.999 − 1.73i)4-s + (4.99 − 0.0121i)5-s + 2.44i·6-s + (0.686 + 6.96i)7-s − 2.82i·8-s + (−1.5 − 2.59i)9-s + (6.11 − 3.55i)10-s + (−2.20 + 3.82i)11-s + (1.73 + 2.99i)12-s + 19.3·13-s + (5.76 + 8.04i)14-s + (−4.31 + 7.51i)15-s + (−2.00 − 3.46i)16-s + (−1.48 + 2.57i)17-s + ⋯ |
| L(s) = 1 | + (0.612 − 0.353i)2-s + (−0.288 + 0.5i)3-s + (0.249 − 0.433i)4-s + (0.999 − 0.00243i)5-s + 0.408i·6-s + (0.0980 + 0.995i)7-s − 0.353i·8-s + (−0.166 − 0.288i)9-s + (0.611 − 0.355i)10-s + (−0.200 + 0.347i)11-s + (0.144 + 0.249i)12-s + 1.49·13-s + (0.411 + 0.574i)14-s + (−0.287 + 0.500i)15-s + (−0.125 − 0.216i)16-s + (−0.0872 + 0.151i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.974 - 0.225i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.974 - 0.225i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.32716 + 0.265688i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.32716 + 0.265688i\) |
| \(L(2)\) |
|
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 | \( 1 + (-4.99 + 0.0121i)T \) |
| 7 | \( 1 + (-0.686 - 6.96i)T \) |
| good | 11 | \( 1 + (2.20 - 3.82i)T + (-60.5 - 104. i)T^{2} \) |
| 13 | \( 1 - 19.3T + 169T^{2} \) |
| 17 | \( 1 + (1.48 - 2.57i)T + (-144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (-2.06 + 1.19i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (1.29 - 0.748i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 - 39.2T + 841T^{2} \) |
| 31 | \( 1 + (35.0 + 20.2i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (36.8 - 21.2i)T + (684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + 57.4iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 53.4iT - 1.84e3T^{2} \) |
| 47 | \( 1 + (38.5 + 66.8i)T + (-1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + (81.3 + 46.9i)T + (1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (8.24 + 4.75i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (35.0 - 20.2i)T + (1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (53.1 + 30.6i)T + (2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 + 46.1T + 5.04e3T^{2} \) |
| 73 | \( 1 + (1.76 - 3.06i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-47.5 - 82.4i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 - 94.2T + 6.88e3T^{2} \) |
| 89 | \( 1 + (-133. + 76.9i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + 122.T + 9.40e3T^{2} \) |
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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
−12.20925336010892969765061983577, −11.19931597752100373258849766649, −10.37881638648109030362412444962, −9.396686149712211198945098831294, −8.490030587206112601060720258942, −6.53102027674399047909270634011, −5.75543566389884370015180879034, −4.84715854160747799933675590765, −3.28934610903465516905829911784, −1.82135127744399885415905101593,
1.41119451324348145536189083293, 3.28066945355850318056040215532, 4.80866427319067818211161236953, 6.00514464327507917144778444654, 6.68865701137411859458595651700, 7.86965258501876419214129425626, 9.004425059148338669716437761142, 10.49310845715749895794702129519, 11.05704938134068806488687817548, 12.38843679966849913382612056667
