| L(s) = 1 | + 2-s + 0.706·3-s + 4-s + 5-s + 0.706·6-s − 4.28·7-s + 8-s − 2.50·9-s + 10-s + 2.56·11-s + 0.706·12-s − 4.03·13-s − 4.28·14-s + 0.706·15-s + 16-s + 5.78·17-s − 2.50·18-s − 1.39·19-s + 20-s − 3.02·21-s + 2.56·22-s + 0.677·23-s + 0.706·24-s + 25-s − 4.03·26-s − 3.88·27-s − 4.28·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.408·3-s + 0.5·4-s + 0.447·5-s + 0.288·6-s − 1.61·7-s + 0.353·8-s − 0.833·9-s + 0.316·10-s + 0.773·11-s + 0.204·12-s − 1.11·13-s − 1.14·14-s + 0.182·15-s + 0.250·16-s + 1.40·17-s − 0.589·18-s − 0.320·19-s + 0.223·20-s − 0.660·21-s + 0.546·22-s + 0.141·23-s + 0.144·24-s + 0.200·25-s − 0.790·26-s − 0.748·27-s − 0.809·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 401 | \( 1 + T \) |
| good | 3 | \( 1 - 0.706T + 3T^{2} \) |
| 7 | \( 1 + 4.28T + 7T^{2} \) |
| 11 | \( 1 - 2.56T + 11T^{2} \) |
| 13 | \( 1 + 4.03T + 13T^{2} \) |
| 17 | \( 1 - 5.78T + 17T^{2} \) |
| 19 | \( 1 + 1.39T + 19T^{2} \) |
| 23 | \( 1 - 0.677T + 23T^{2} \) |
| 29 | \( 1 + 4.24T + 29T^{2} \) |
| 31 | \( 1 + 6.43T + 31T^{2} \) |
| 37 | \( 1 + 4.62T + 37T^{2} \) |
| 41 | \( 1 + 9.36T + 41T^{2} \) |
| 43 | \( 1 - 3.25T + 43T^{2} \) |
| 47 | \( 1 + 0.866T + 47T^{2} \) |
| 53 | \( 1 + 3.71T + 53T^{2} \) |
| 59 | \( 1 + 4.31T + 59T^{2} \) |
| 61 | \( 1 - 9.20T + 61T^{2} \) |
| 67 | \( 1 + 13.7T + 67T^{2} \) |
| 71 | \( 1 - 1.82T + 71T^{2} \) |
| 73 | \( 1 + 9.24T + 73T^{2} \) |
| 79 | \( 1 + 4.95T + 79T^{2} \) |
| 83 | \( 1 - 10.4T + 83T^{2} \) |
| 89 | \( 1 + 3.92T + 89T^{2} \) |
| 97 | \( 1 + 8.02T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.976903956156646884572097606277, −7.11614182823006039177933165860, −6.59492320324223084807535918677, −5.71085910613356677234935787973, −5.34240801858587366880460662947, −4.03683571828830901649718199407, −3.29375456169382467853291312091, −2.82707248507748980088446471343, −1.73304216701051429477529393921, 0,
1.73304216701051429477529393921, 2.82707248507748980088446471343, 3.29375456169382467853291312091, 4.03683571828830901649718199407, 5.34240801858587366880460662947, 5.71085910613356677234935787973, 6.59492320324223084807535918677, 7.11614182823006039177933165860, 7.976903956156646884572097606277
