L(s) = 1 | + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s + (−0.5 − 0.866i)5-s + 3·7-s − 0.999·8-s + (0.499 − 0.866i)10-s − 11-s + (−1 + 1.73i)13-s + (1.5 + 2.59i)14-s + (−0.5 − 0.866i)16-s + (1 + 1.73i)17-s + (0.5 + 4.33i)19-s + 0.999·20-s + (−0.5 − 0.866i)22-s + (0.5 − 0.866i)23-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.223 − 0.387i)5-s + 1.13·7-s − 0.353·8-s + (0.158 − 0.273i)10-s − 0.301·11-s + (−0.277 + 0.480i)13-s + (0.400 + 0.694i)14-s + (−0.125 − 0.216i)16-s + (0.242 + 0.420i)17-s + (0.114 + 0.993i)19-s + 0.223·20-s + (−0.106 − 0.184i)22-s + (0.104 − 0.180i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0977 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0977 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.069666038\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.069666038\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.5 - 4.33i)T \) |
good | 7 | \( 1 - 3T + 7T^{2} \) |
| 11 | \( 1 + T + 11T^{2} \) |
| 13 | \( 1 + (1 - 1.73i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-1 - 1.73i)T + (-8.5 + 14.7i)T^{2} \) |
| 23 | \( 1 + (-0.5 + 0.866i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 - 9T + 37T^{2} \) |
| 41 | \( 1 + (-0.5 - 0.866i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-5 - 8.66i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-1.5 + 2.59i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-6 - 10.3i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-1 + 1.73i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-1 + 1.73i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-4 - 6.92i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-6 - 10.3i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (7 + 12.1i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 6T + 83T^{2} \) |
| 89 | \( 1 + (2.5 - 4.33i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (4 + 6.92i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.389156539245316422905992646167, −8.349408563389744036958359762603, −8.029103554375264075708749801953, −7.25361188835612940501957004921, −6.19152075228147021044937933238, −5.42809001240074574679078417707, −4.58309503670190111370410861752, −3.99593126082916570078833286023, −2.60840458665968772588756664149, −1.28234868870439648036751210380,
0.791379575076051957947039970828, 2.20861908644391268531740518711, 2.99579366212051413716056610048, 4.15940121884563160129347448406, 4.94429593282086487103943385517, 5.61959985657414192425337522229, 6.79153371755695353898350371960, 7.65268048739800356716683209815, 8.301522288546477268076153838221, 9.280851674372576138784793775995