| L(s) = 1 | − 2-s + 2.53·3-s + 4-s + 5-s − 2.53·6-s − 0.583·7-s − 8-s + 3.41·9-s − 10-s + 11-s + 2.53·12-s − 4.55·13-s + 0.583·14-s + 2.53·15-s + 16-s + 5.04·17-s − 3.41·18-s − 4.92·19-s + 20-s − 1.47·21-s − 22-s − 0.815·23-s − 2.53·24-s + 25-s + 4.55·26-s + 1.04·27-s − 0.583·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.46·3-s + 0.5·4-s + 0.447·5-s − 1.03·6-s − 0.220·7-s − 0.353·8-s + 1.13·9-s − 0.316·10-s + 0.301·11-s + 0.731·12-s − 1.26·13-s + 0.156·14-s + 0.653·15-s + 0.250·16-s + 1.22·17-s − 0.804·18-s − 1.13·19-s + 0.223·20-s − 0.322·21-s − 0.213·22-s − 0.170·23-s − 0.516·24-s + 0.200·25-s + 0.893·26-s + 0.201·27-s − 0.110·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.558089052\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.558089052\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 43 | \( 1 - T \) |
| good | 3 | \( 1 - 2.53T + 3T^{2} \) |
| 7 | \( 1 + 0.583T + 7T^{2} \) |
| 13 | \( 1 + 4.55T + 13T^{2} \) |
| 17 | \( 1 - 5.04T + 17T^{2} \) |
| 19 | \( 1 + 4.92T + 19T^{2} \) |
| 23 | \( 1 + 0.815T + 23T^{2} \) |
| 29 | \( 1 - 7.94T + 29T^{2} \) |
| 31 | \( 1 + 2.45T + 31T^{2} \) |
| 37 | \( 1 - 3.39T + 37T^{2} \) |
| 41 | \( 1 - 10.9T + 41T^{2} \) |
| 47 | \( 1 - 7.21T + 47T^{2} \) |
| 53 | \( 1 - 11.9T + 53T^{2} \) |
| 59 | \( 1 - 9.88T + 59T^{2} \) |
| 61 | \( 1 + 7.37T + 61T^{2} \) |
| 67 | \( 1 - 9.44T + 67T^{2} \) |
| 71 | \( 1 - 11.2T + 71T^{2} \) |
| 73 | \( 1 + 1.95T + 73T^{2} \) |
| 79 | \( 1 - 16.0T + 79T^{2} \) |
| 83 | \( 1 - 0.0176T + 83T^{2} \) |
| 89 | \( 1 + 4.04T + 89T^{2} \) |
| 97 | \( 1 - 3.09T + 97T^{2} \) |
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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
−8.305373876005559349947353137368, −7.78901787421340157258402551832, −7.12554467773748460824184789746, −6.35639974354858059893731134090, −5.45010113724764561206571555981, −4.36520633922232824679402867106, −3.51020198994313868355068348044, −2.53958591765431849897835879777, −2.22614517390393884850872704047, −0.920448080193192325138220393818,
0.920448080193192325138220393818, 2.22614517390393884850872704047, 2.53958591765431849897835879777, 3.51020198994313868355068348044, 4.36520633922232824679402867106, 5.45010113724764561206571555981, 6.35639974354858059893731134090, 7.12554467773748460824184789746, 7.78901787421340157258402551832, 8.305373876005559349947353137368
