| L(s) = 1 | − 2.66·2-s − 3-s + 5.09·4-s + 2.66·6-s + 0.0170·7-s − 8.24·8-s + 9-s + 1.70·11-s − 5.09·12-s + 4.69·13-s − 0.0453·14-s + 11.7·16-s + 3.91·17-s − 2.66·18-s + 6.02·19-s − 0.0170·21-s − 4.55·22-s − 4.82·23-s + 8.24·24-s − 12.5·26-s − 27-s + 0.0868·28-s − 29-s + 1.33·31-s − 14.8·32-s − 1.70·33-s − 10.4·34-s + ⋯ |
| L(s) = 1 | − 1.88·2-s − 0.577·3-s + 2.54·4-s + 1.08·6-s + 0.00644·7-s − 2.91·8-s + 0.333·9-s + 0.515·11-s − 1.47·12-s + 1.30·13-s − 0.0121·14-s + 2.94·16-s + 0.949·17-s − 0.627·18-s + 1.38·19-s − 0.00371·21-s − 0.971·22-s − 1.00·23-s + 1.68·24-s − 2.45·26-s − 0.192·27-s + 0.0164·28-s − 0.185·29-s + 0.240·31-s − 2.62·32-s − 0.297·33-s − 1.78·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7213963684\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7213963684\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 + 2.66T + 2T^{2} \) |
| 7 | \( 1 - 0.0170T + 7T^{2} \) |
| 11 | \( 1 - 1.70T + 11T^{2} \) |
| 13 | \( 1 - 4.69T + 13T^{2} \) |
| 17 | \( 1 - 3.91T + 17T^{2} \) |
| 19 | \( 1 - 6.02T + 19T^{2} \) |
| 23 | \( 1 + 4.82T + 23T^{2} \) |
| 31 | \( 1 - 1.33T + 31T^{2} \) |
| 37 | \( 1 - 8.18T + 37T^{2} \) |
| 41 | \( 1 - 8.36T + 41T^{2} \) |
| 43 | \( 1 + 3.72T + 43T^{2} \) |
| 47 | \( 1 - 5.36T + 47T^{2} \) |
| 53 | \( 1 - 7.21T + 53T^{2} \) |
| 59 | \( 1 + 8.65T + 59T^{2} \) |
| 61 | \( 1 - 5.72T + 61T^{2} \) |
| 67 | \( 1 + 3.87T + 67T^{2} \) |
| 71 | \( 1 + 4.32T + 71T^{2} \) |
| 73 | \( 1 - 1.78T + 73T^{2} \) |
| 79 | \( 1 - 0.233T + 79T^{2} \) |
| 83 | \( 1 - 6.15T + 83T^{2} \) |
| 89 | \( 1 + 12.4T + 89T^{2} \) |
| 97 | \( 1 - 19.2T + 97T^{2} \) |
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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
−9.304905015397407057984508886823, −8.218152379239932118776538293800, −7.78918940137861932611097369539, −6.94241326170045472754322790506, −6.12937338899109650037635792537, −5.60728342299222017101273006059, −3.98402738369101836133915587932, −2.90763899115558260939117157652, −1.53708590639601758346834133363, −0.834733188292331876571569817971,
0.834733188292331876571569817971, 1.53708590639601758346834133363, 2.90763899115558260939117157652, 3.98402738369101836133915587932, 5.60728342299222017101273006059, 6.12937338899109650037635792537, 6.94241326170045472754322790506, 7.78918940137861932611097369539, 8.218152379239932118776538293800, 9.304905015397407057984508886823
