| L(s) = 1 | + 2-s − 1.49·3-s + 4-s − 5-s − 1.49·6-s − 7-s + 8-s − 0.770·9-s − 10-s − 3.24·11-s − 1.49·12-s + 5.86·13-s − 14-s + 1.49·15-s + 16-s − 5.04·17-s − 0.770·18-s + 1.12·19-s − 20-s + 1.49·21-s − 3.24·22-s + 6.84·23-s − 1.49·24-s + 25-s + 5.86·26-s + 5.62·27-s − 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.862·3-s + 0.5·4-s − 0.447·5-s − 0.609·6-s − 0.377·7-s + 0.353·8-s − 0.256·9-s − 0.316·10-s − 0.978·11-s − 0.431·12-s + 1.62·13-s − 0.267·14-s + 0.385·15-s + 0.250·16-s − 1.22·17-s − 0.181·18-s + 0.257·19-s − 0.223·20-s + 0.325·21-s − 0.692·22-s + 1.42·23-s − 0.304·24-s + 0.200·25-s + 1.15·26-s + 1.08·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4970 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4970 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 71 | \( 1 - T \) |
| good | 3 | \( 1 + 1.49T + 3T^{2} \) |
| 11 | \( 1 + 3.24T + 11T^{2} \) |
| 13 | \( 1 - 5.86T + 13T^{2} \) |
| 17 | \( 1 + 5.04T + 17T^{2} \) |
| 19 | \( 1 - 1.12T + 19T^{2} \) |
| 23 | \( 1 - 6.84T + 23T^{2} \) |
| 29 | \( 1 - 4.15T + 29T^{2} \) |
| 31 | \( 1 + 1.02T + 31T^{2} \) |
| 37 | \( 1 + 6.05T + 37T^{2} \) |
| 41 | \( 1 - 5.54T + 41T^{2} \) |
| 43 | \( 1 - 2.89T + 43T^{2} \) |
| 47 | \( 1 + 7.93T + 47T^{2} \) |
| 53 | \( 1 + 8.35T + 53T^{2} \) |
| 59 | \( 1 - 2.72T + 59T^{2} \) |
| 61 | \( 1 + 5.61T + 61T^{2} \) |
| 67 | \( 1 + 8.43T + 67T^{2} \) |
| 73 | \( 1 + 4.54T + 73T^{2} \) |
| 79 | \( 1 + 11.8T + 79T^{2} \) |
| 83 | \( 1 + 5.16T + 83T^{2} \) |
| 89 | \( 1 - 15.1T + 89T^{2} \) |
| 97 | \( 1 + 13.2T + 97T^{2} \) |
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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
−7.78245183545858775372444133199, −6.83724007424459178453169872402, −6.38587733379585026961924708795, −5.66820530299764898071038944733, −4.99368580591439088648874331468, −4.31109788268496712272435104857, −3.30410795871667282010206013687, −2.71762873051039933190187524981, −1.26762665413203278989207382941, 0,
1.26762665413203278989207382941, 2.71762873051039933190187524981, 3.30410795871667282010206013687, 4.31109788268496712272435104857, 4.99368580591439088648874331468, 5.66820530299764898071038944733, 6.38587733379585026961924708795, 6.83724007424459178453169872402, 7.78245183545858775372444133199
