L(s) = 1 | + (0.618 − 1.90i)2-s + 2.35i·3-s + (−3.23 − 2.35i)4-s − 2.23·5-s + (4.47 + 1.45i)6-s + 5.25i·7-s + (−6.47 + 4.70i)8-s + 3.47·9-s + (−1.38 + 4.25i)10-s − 19.9i·11-s + (5.52 − 7.60i)12-s − 8.47·13-s + (9.99 + 3.24i)14-s − 5.25i·15-s + (4.94 + 15.2i)16-s + 11.8·17-s + ⋯ |
L(s) = 1 | + (0.309 − 0.951i)2-s + 0.783i·3-s + (−0.809 − 0.587i)4-s − 0.447·5-s + (0.745 + 0.242i)6-s + 0.751i·7-s + (−0.809 + 0.587i)8-s + 0.385·9-s + (−0.138 + 0.425i)10-s − 1.81i·11-s + (0.460 − 0.634i)12-s − 0.651·13-s + (0.714 + 0.232i)14-s − 0.350i·15-s + (0.309 + 0.951i)16-s + 0.699·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.809 + 0.587i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.809 + 0.587i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.850569 - 0.276366i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.850569 - 0.276366i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.618 + 1.90i)T \) |
| 5 | \( 1 + 2.23T \) |
good | 3 | \( 1 - 2.35iT - 9T^{2} \) |
| 7 | \( 1 - 5.25iT - 49T^{2} \) |
| 11 | \( 1 + 19.9iT - 121T^{2} \) |
| 13 | \( 1 + 8.47T + 169T^{2} \) |
| 17 | \( 1 - 11.8T + 289T^{2} \) |
| 19 | \( 1 - 15.2iT - 361T^{2} \) |
| 23 | \( 1 - 0.555iT - 529T^{2} \) |
| 29 | \( 1 + 10.9T + 841T^{2} \) |
| 31 | \( 1 - 8.29iT - 961T^{2} \) |
| 37 | \( 1 + 18.3T + 1.36e3T^{2} \) |
| 41 | \( 1 + 14.5T + 1.68e3T^{2} \) |
| 43 | \( 1 + 22.2iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 53.3iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 66.3T + 2.80e3T^{2} \) |
| 59 | \( 1 + 17.4iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 90.1T + 3.72e3T^{2} \) |
| 67 | \( 1 + 50.2iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 80.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 5.55T + 5.32e3T^{2} \) |
| 79 | \( 1 - 13.8iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 76.2iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 111.T + 7.92e3T^{2} \) |
| 97 | \( 1 + 92.8T + 9.40e3T^{2} \) |
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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
−18.58585532578778796125086203024, −16.60144260466840259826373441057, −15.34922760887369189913514808858, −14.11965764076519389585473530947, −12.49875361328906704754609653084, −11.29415588817075777185017400195, −10.01272837542119754235070824034, −8.592821475085787209603572063168, −5.43441430703993799926017740945, −3.53233895242241329213728963533,
4.53806032610858065508113104593, 6.98214116978119855428849482800, 7.64047041750971698148875835846, 9.788602370359588837916928515174, 12.21303170706440393411827475958, 13.08663694101274076841807556316, 14.51309729259349045552606087342, 15.62779135732896705591601604435, 17.10977597074751866119696618412, 17.96639225779597393254419667225