| L(s) = 1 | + 2-s + 2.68·3-s + 4-s − 5-s + 2.68·6-s − 4.59·7-s + 8-s + 4.22·9-s − 10-s + 5.13·11-s + 2.68·12-s − 1.22·13-s − 4.59·14-s − 2.68·15-s + 16-s − 4.68·17-s + 4.22·18-s − 4.59·19-s − 20-s − 12.3·21-s + 5.13·22-s − 23-s + 2.68·24-s + 25-s − 1.22·26-s + 3.28·27-s − 4.59·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.55·3-s + 0.5·4-s − 0.447·5-s + 1.09·6-s − 1.73·7-s + 0.353·8-s + 1.40·9-s − 0.316·10-s + 1.54·11-s + 0.775·12-s − 0.338·13-s − 1.22·14-s − 0.693·15-s + 0.250·16-s − 1.13·17-s + 0.995·18-s − 1.05·19-s − 0.223·20-s − 2.69·21-s + 1.09·22-s − 0.208·23-s + 0.548·24-s + 0.200·25-s − 0.239·26-s + 0.632·27-s − 0.868·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.359869302\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.359869302\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 - 2.68T + 3T^{2} \) |
| 7 | \( 1 + 4.59T + 7T^{2} \) |
| 11 | \( 1 - 5.13T + 11T^{2} \) |
| 13 | \( 1 + 1.22T + 13T^{2} \) |
| 17 | \( 1 + 4.68T + 17T^{2} \) |
| 19 | \( 1 + 4.59T + 19T^{2} \) |
| 29 | \( 1 - 3.37T + 29T^{2} \) |
| 31 | \( 1 + 0.777T + 31T^{2} \) |
| 37 | \( 1 - 5.81T + 37T^{2} \) |
| 41 | \( 1 + 8.50T + 41T^{2} \) |
| 43 | \( 1 - 8T + 43T^{2} \) |
| 47 | \( 1 + 6.44T + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 - 9.37T + 59T^{2} \) |
| 61 | \( 1 - 10.9T + 61T^{2} \) |
| 67 | \( 1 - 15.6T + 67T^{2} \) |
| 71 | \( 1 - 1.31T + 71T^{2} \) |
| 73 | \( 1 + 4.44T + 73T^{2} \) |
| 79 | \( 1 + 4.88T + 79T^{2} \) |
| 83 | \( 1 + 3.81T + 83T^{2} \) |
| 89 | \( 1 - 8.93T + 89T^{2} \) |
| 97 | \( 1 + 18.0T + 97T^{2} \) |
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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
−12.64051517333045021380319705344, −11.47471728390542385508531428870, −10.01054966463209336887680512748, −9.221193891353278168147803757554, −8.396012933536110344209181167679, −6.95349431963623971899874497637, −6.41585253670014841738370349535, −4.21837372826066387075583667493, −3.54703003231268187357050421861, −2.41708781421801947228034811566,
2.41708781421801947228034811566, 3.54703003231268187357050421861, 4.21837372826066387075583667493, 6.41585253670014841738370349535, 6.95349431963623971899874497637, 8.396012933536110344209181167679, 9.221193891353278168147803757554, 10.01054966463209336887680512748, 11.47471728390542385508531428870, 12.64051517333045021380319705344
