L(s) = 1 | + (−0.5 − 0.866i)2-s + (−0.5 + 0.866i)3-s + (−0.499 + 0.866i)4-s + (−1 − 1.73i)5-s + 0.999·6-s + 0.999·8-s + (−0.499 − 0.866i)9-s + (−0.999 + 1.73i)10-s + (2 − 3.46i)11-s + (−0.499 − 0.866i)12-s − 6·13-s + 1.99·15-s + (−0.5 − 0.866i)16-s + (1 − 1.73i)17-s + (−0.499 + 0.866i)18-s + (−2 − 3.46i)19-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (−0.288 + 0.499i)3-s + (−0.249 + 0.433i)4-s + (−0.447 − 0.774i)5-s + 0.408·6-s + 0.353·8-s + (−0.166 − 0.288i)9-s + (−0.316 + 0.547i)10-s + (0.603 − 1.04i)11-s + (−0.144 − 0.249i)12-s − 1.66·13-s + 0.516·15-s + (−0.125 − 0.216i)16-s + (0.242 − 0.420i)17-s + (−0.117 + 0.204i)18-s + (−0.458 − 0.794i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.701 + 0.712i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.701 + 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.230379 - 0.549722i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.230379 - 0.549722i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.5 - 0.866i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (1 + 1.73i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-2 + 3.46i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 6T + 13T^{2} \) |
| 17 | \( 1 + (-1 + 1.73i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (2 + 3.46i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (4 + 6.92i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-5 - 8.66i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (3 - 5.19i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-2 + 3.46i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-3 - 5.19i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (2 - 3.46i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 + (-5 + 8.66i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 4T + 83T^{2} \) |
| 89 | \( 1 + (3 + 5.19i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 14T + 97T^{2} \) |
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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
−11.52139013956716966692339485737, −10.48893098039013274090130252498, −9.571322768419506760571051977178, −8.766647804035256629565796305818, −7.88157836693051664150520105718, −6.47275686945982252098654723090, −4.97636167220636002120561114835, −4.21536856473896139180963478796, −2.71582742669109014304947578532, −0.50536055101104460612865375100,
2.04072986579305853023600278227, 3.93988941373864831048355907815, 5.32582021709298821923335436848, 6.49004749384355708572760411240, 7.40517947442939017957074667149, 7.80714630839587868580542486676, 9.408518829385937771906293456981, 10.08241260517181695135022430656, 11.21874463868288977456539625125, 12.13088943882806020005347097349