L(s) = 1 | + 2.44·2-s − 2.95·3-s + 3.95·4-s + 5-s − 7.21·6-s − 2.50·7-s + 4.78·8-s + 5.72·9-s + 2.44·10-s + 11-s − 11.6·12-s + 4.03·13-s − 6.11·14-s − 2.95·15-s + 3.75·16-s − 4.47·17-s + 13.9·18-s + 2.04·19-s + 3.95·20-s + 7.40·21-s + 2.44·22-s − 5.54·23-s − 14.1·24-s + 25-s + 9.84·26-s − 8.05·27-s − 9.91·28-s + ⋯ |
L(s) = 1 | + 1.72·2-s − 1.70·3-s + 1.97·4-s + 0.447·5-s − 2.94·6-s − 0.946·7-s + 1.69·8-s + 1.90·9-s + 0.771·10-s + 0.301·11-s − 3.37·12-s + 1.11·13-s − 1.63·14-s − 0.762·15-s + 0.938·16-s − 1.08·17-s + 3.29·18-s + 0.468·19-s + 0.885·20-s + 1.61·21-s + 0.520·22-s − 1.15·23-s − 2.88·24-s + 0.200·25-s + 1.93·26-s − 1.55·27-s − 1.87·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.140513407\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.140513407\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 73 | \( 1 - T \) |
good | 2 | \( 1 - 2.44T + 2T^{2} \) |
| 3 | \( 1 + 2.95T + 3T^{2} \) |
| 7 | \( 1 + 2.50T + 7T^{2} \) |
| 13 | \( 1 - 4.03T + 13T^{2} \) |
| 17 | \( 1 + 4.47T + 17T^{2} \) |
| 19 | \( 1 - 2.04T + 19T^{2} \) |
| 23 | \( 1 + 5.54T + 23T^{2} \) |
| 29 | \( 1 - 6.07T + 29T^{2} \) |
| 31 | \( 1 - 8.54T + 31T^{2} \) |
| 37 | \( 1 - 4.41T + 37T^{2} \) |
| 41 | \( 1 - 4.39T + 41T^{2} \) |
| 43 | \( 1 + 6.31T + 43T^{2} \) |
| 47 | \( 1 - 5.44T + 47T^{2} \) |
| 53 | \( 1 - 0.170T + 53T^{2} \) |
| 59 | \( 1 + 0.182T + 59T^{2} \) |
| 61 | \( 1 - 8.80T + 61T^{2} \) |
| 67 | \( 1 - 3.65T + 67T^{2} \) |
| 71 | \( 1 - 2.71T + 71T^{2} \) |
| 79 | \( 1 - 4.63T + 79T^{2} \) |
| 83 | \( 1 - 10.8T + 83T^{2} \) |
| 89 | \( 1 - 5.44T + 89T^{2} \) |
| 97 | \( 1 - 0.364T + 97T^{2} \) |
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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
−8.266743853902802629107733024082, −6.94367924071827056257345417429, −6.47582037014665066396357820445, −6.15138609351027696154103336629, −5.58956308205001602321512306277, −4.69279931744594129446608606537, −4.18605832583150291246569648458, −3.28255280156726368423821722248, −2.18287878016278950923850506794, −0.858706378796804429980702497902,
0.858706378796804429980702497902, 2.18287878016278950923850506794, 3.28255280156726368423821722248, 4.18605832583150291246569648458, 4.69279931744594129446608606537, 5.58956308205001602321512306277, 6.15138609351027696154103336629, 6.47582037014665066396357820445, 6.94367924071827056257345417429, 8.266743853902802629107733024082