L(s) = 1 | − 2.21·2-s − 2.59·3-s + 2.91·4-s + 1.00·5-s + 5.74·6-s − 1.82·7-s − 2.02·8-s + 3.71·9-s − 2.22·10-s − 2.73·11-s − 7.55·12-s + 4.03·14-s − 2.60·15-s − 1.33·16-s − 2.66·17-s − 8.24·18-s + 2.04·19-s + 2.93·20-s + 4.71·21-s + 6.06·22-s − 1.36·23-s + 5.25·24-s − 3.98·25-s − 1.85·27-s − 5.30·28-s − 3.75·29-s + 5.77·30-s + ⋯ |
L(s) = 1 | − 1.56·2-s − 1.49·3-s + 1.45·4-s + 0.449·5-s + 2.34·6-s − 0.688·7-s − 0.716·8-s + 1.23·9-s − 0.705·10-s − 0.824·11-s − 2.18·12-s + 1.07·14-s − 0.673·15-s − 0.334·16-s − 0.647·17-s − 1.94·18-s + 0.469·19-s + 0.655·20-s + 1.02·21-s + 1.29·22-s − 0.283·23-s + 1.07·24-s − 0.797·25-s − 0.357·27-s − 1.00·28-s − 0.696·29-s + 1.05·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5239 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5239 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1059889019\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1059889019\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 13 | \( 1 \) |
| 31 | \( 1 + T \) |
good | 2 | \( 1 + 2.21T + 2T^{2} \) |
| 3 | \( 1 + 2.59T + 3T^{2} \) |
| 5 | \( 1 - 1.00T + 5T^{2} \) |
| 7 | \( 1 + 1.82T + 7T^{2} \) |
| 11 | \( 1 + 2.73T + 11T^{2} \) |
| 17 | \( 1 + 2.66T + 17T^{2} \) |
| 19 | \( 1 - 2.04T + 19T^{2} \) |
| 23 | \( 1 + 1.36T + 23T^{2} \) |
| 29 | \( 1 + 3.75T + 29T^{2} \) |
| 37 | \( 1 + 5.04T + 37T^{2} \) |
| 41 | \( 1 + 3.02T + 41T^{2} \) |
| 43 | \( 1 + 9.12T + 43T^{2} \) |
| 47 | \( 1 + 7.62T + 47T^{2} \) |
| 53 | \( 1 - 5.26T + 53T^{2} \) |
| 59 | \( 1 + 2.66T + 59T^{2} \) |
| 61 | \( 1 - 7.14T + 61T^{2} \) |
| 67 | \( 1 - 9.48T + 67T^{2} \) |
| 71 | \( 1 - 4.99T + 71T^{2} \) |
| 73 | \( 1 + 4.50T + 73T^{2} \) |
| 79 | \( 1 + 0.813T + 79T^{2} \) |
| 83 | \( 1 + 1.18T + 83T^{2} \) |
| 89 | \( 1 - 16.2T + 89T^{2} \) |
| 97 | \( 1 + 3.69T + 97T^{2} \) |
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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
−8.226589698882379197341452822951, −7.49760054823191164389749170829, −6.71023692475335425481751710792, −6.35511628733672379043796007486, −5.44040402172472041871510984923, −4.90954734418192680864858929085, −3.62564482059638851171649433049, −2.36332041273623034655983968621, −1.48260718034285220702445510520, −0.24604787627335026792293288491,
0.24604787627335026792293288491, 1.48260718034285220702445510520, 2.36332041273623034655983968621, 3.62564482059638851171649433049, 4.90954734418192680864858929085, 5.44040402172472041871510984923, 6.35511628733672379043796007486, 6.71023692475335425481751710792, 7.49760054823191164389749170829, 8.226589698882379197341452822951