| L(s) = 1 | − 924.·3-s + 2.37e5·7-s − 7.39e5·9-s + 4.57e6·11-s − 1.47e7·13-s − 1.18e8·17-s − 3.43e8·19-s − 2.19e8·21-s + 7.20e8·23-s + 2.15e9·27-s + 3.84e9·29-s + 1.88e9·31-s − 4.23e9·33-s − 1.25e10·37-s + 1.36e10·39-s + 3.99e9·41-s − 4.09e10·43-s + 1.14e11·47-s − 4.02e10·49-s + 1.09e11·51-s − 2.02e11·53-s + 3.17e11·57-s + 8.87e10·59-s − 5.08e11·61-s − 1.75e11·63-s + 4.62e11·67-s − 6.66e11·69-s + ⋯ |
| L(s) = 1 | − 0.732·3-s + 0.764·7-s − 0.463·9-s + 0.779·11-s − 0.847·13-s − 1.18·17-s − 1.67·19-s − 0.559·21-s + 1.01·23-s + 1.07·27-s + 1.20·29-s + 0.380·31-s − 0.570·33-s − 0.804·37-s + 0.620·39-s + 0.131·41-s − 0.988·43-s + 1.54·47-s − 0.415·49-s + 0.868·51-s − 1.25·53-s + 1.22·57-s + 0.273·59-s − 1.26·61-s − 0.354·63-s + 0.625·67-s − 0.743·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(1.120506766\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.120506766\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 924.T + 1.59e6T^{2} \) |
| 7 | \( 1 - 2.37e5T + 9.68e10T^{2} \) |
| 11 | \( 1 - 4.57e6T + 3.45e13T^{2} \) |
| 13 | \( 1 + 1.47e7T + 3.02e14T^{2} \) |
| 17 | \( 1 + 1.18e8T + 9.90e15T^{2} \) |
| 19 | \( 1 + 3.43e8T + 4.20e16T^{2} \) |
| 23 | \( 1 - 7.20e8T + 5.04e17T^{2} \) |
| 29 | \( 1 - 3.84e9T + 1.02e19T^{2} \) |
| 31 | \( 1 - 1.88e9T + 2.44e19T^{2} \) |
| 37 | \( 1 + 1.25e10T + 2.43e20T^{2} \) |
| 41 | \( 1 - 3.99e9T + 9.25e20T^{2} \) |
| 43 | \( 1 + 4.09e10T + 1.71e21T^{2} \) |
| 47 | \( 1 - 1.14e11T + 5.46e21T^{2} \) |
| 53 | \( 1 + 2.02e11T + 2.60e22T^{2} \) |
| 59 | \( 1 - 8.87e10T + 1.04e23T^{2} \) |
| 61 | \( 1 + 5.08e11T + 1.61e23T^{2} \) |
| 67 | \( 1 - 4.62e11T + 5.48e23T^{2} \) |
| 71 | \( 1 + 3.98e10T + 1.16e24T^{2} \) |
| 73 | \( 1 + 1.11e12T + 1.67e24T^{2} \) |
| 79 | \( 1 + 2.87e12T + 4.66e24T^{2} \) |
| 83 | \( 1 - 1.85e12T + 8.87e24T^{2} \) |
| 89 | \( 1 - 4.00e12T + 2.19e25T^{2} \) |
| 97 | \( 1 + 1.06e13T + 6.73e25T^{2} \) |
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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
−10.43313706561481960591903234191, −9.021276746920388963208650902616, −8.343486822364179397526532754169, −6.93009850684739976333727320673, −6.23929567672328596610671564023, −4.96912023675457443029975885231, −4.37127777795380698901866748677, −2.76660687567541943570122158338, −1.68531443861349284172231718435, −0.45139452367579231556645716857,
0.45139452367579231556645716857, 1.68531443861349284172231718435, 2.76660687567541943570122158338, 4.37127777795380698901866748677, 4.96912023675457443029975885231, 6.23929567672328596610671564023, 6.93009850684739976333727320673, 8.343486822364179397526532754169, 9.021276746920388963208650902616, 10.43313706561481960591903234191
