| L(s) = 1 | − 2-s − 2·3-s + 4-s − 2·5-s + 2·6-s + 7-s − 8-s + 9-s + 2·10-s − 6·11-s − 2·12-s − 4·13-s − 14-s + 4·15-s + 16-s + 2·17-s − 18-s − 4·19-s − 2·20-s − 2·21-s + 6·22-s + 23-s + 2·24-s − 25-s + 4·26-s + 4·27-s + 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s + 1/2·4-s − 0.894·5-s + 0.816·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.632·10-s − 1.80·11-s − 0.577·12-s − 1.10·13-s − 0.267·14-s + 1.03·15-s + 1/4·16-s + 0.485·17-s − 0.235·18-s − 0.917·19-s − 0.447·20-s − 0.436·21-s + 1.27·22-s + 0.208·23-s + 0.408·24-s − 1/5·25-s + 0.784·26-s + 0.769·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 270802 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 270802 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−12.76171501336546, −12.41631472836800, −11.95679953946480, −11.48129751103156, −11.26731616860684, −10.72819946238592, −10.23437780777451, −10.01998378947686, −9.490757956168060, −8.592329306852397, −8.260636493521090, −7.876096767971233, −7.545720257626819, −6.953771794654649, −6.430606130412660, −5.944004904387205, −5.321407104917694, −4.914090504093759, −4.620900875440099, −3.867798470529452, −3.085553751026631, −2.541134604256920, −2.150336272968780, −1.115620048941077, −0.4818024800287352, 0,
0.4818024800287352, 1.115620048941077, 2.150336272968780, 2.541134604256920, 3.085553751026631, 3.867798470529452, 4.620900875440099, 4.914090504093759, 5.321407104917694, 5.944004904387205, 6.430606130412660, 6.953771794654649, 7.545720257626819, 7.876096767971233, 8.260636493521090, 8.592329306852397, 9.490757956168060, 10.01998378947686, 10.23437780777451, 10.72819946238592, 11.26731616860684, 11.48129751103156, 11.95679953946480, 12.41631472836800, 12.76171501336546
