| L(s) = 1 | + 13.4·2-s + 27·3-s + 52.1·4-s − 246.·5-s + 362.·6-s + 343·7-s − 1.01e3·8-s + 729·9-s − 3.30e3·10-s + 8.37e3·11-s + 1.40e3·12-s + 1.01e4·13-s + 4.60e3·14-s − 6.64e3·15-s − 2.03e4·16-s − 2.03e4·17-s + 9.78e3·18-s − 3.86e4·19-s − 1.28e4·20-s + 9.26e3·21-s + 1.12e5·22-s − 1.21e4·23-s − 2.74e4·24-s − 1.75e4·25-s + 1.36e5·26-s + 1.96e4·27-s + 1.79e4·28-s + ⋯ |
| L(s) = 1 | + 1.18·2-s + 0.577·3-s + 0.407·4-s − 0.880·5-s + 0.685·6-s + 0.377·7-s − 0.702·8-s + 0.333·9-s − 1.04·10-s + 1.89·11-s + 0.235·12-s + 1.28·13-s + 0.448·14-s − 0.508·15-s − 1.24·16-s − 1.00·17-s + 0.395·18-s − 1.29·19-s − 0.359·20-s + 0.218·21-s + 2.25·22-s − 0.208·23-s − 0.405·24-s − 0.224·25-s + 1.52·26-s + 0.192·27-s + 0.154·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(4.897100242\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.897100242\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 27T \) |
| 7 | \( 1 - 343T \) |
| 23 | \( 1 + 1.21e4T \) |
| good | 2 | \( 1 - 13.4T + 128T^{2} \) |
| 5 | \( 1 + 246.T + 7.81e4T^{2} \) |
| 11 | \( 1 - 8.37e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 1.01e4T + 6.27e7T^{2} \) |
| 17 | \( 1 + 2.03e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 3.86e4T + 8.93e8T^{2} \) |
| 29 | \( 1 - 1.58e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.64e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 3.33e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 2.54e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 7.80e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 2.54e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.76e6T + 1.17e12T^{2} \) |
| 59 | \( 1 - 2.33e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 2.75e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 3.70e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 4.16e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 5.41e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 3.87e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 4.16e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 1.74e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.63e7T + 8.07e13T^{2} \) |
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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
−9.713602840233318637038935235677, −8.595087441869655397059906328906, −8.343000493388959922487582683218, −6.62601454922063284753406173504, −6.33810092203924181860144688681, −4.66229717366146075337992945656, −4.05405391349074319921619822823, −3.52870039779069291769934206957, −2.14821789919257071501164223164, −0.813214423138342403876150160474,
0.813214423138342403876150160474, 2.14821789919257071501164223164, 3.52870039779069291769934206957, 4.05405391349074319921619822823, 4.66229717366146075337992945656, 6.33810092203924181860144688681, 6.62601454922063284753406173504, 8.343000493388959922487582683218, 8.595087441869655397059906328906, 9.713602840233318637038935235677
