| L(s) = 1 | − 44.4·2-s + 1.46e3·4-s + 1.05e3·5-s + 6.59e3·7-s − 4.24e4·8-s − 4.67e4·10-s − 1.44e4·11-s + 5.00e3·13-s − 2.93e5·14-s + 1.13e6·16-s + 4.44e5·17-s − 4.87e5·19-s + 1.54e6·20-s + 6.43e5·22-s + 6.73e4·23-s − 8.49e5·25-s − 2.22e5·26-s + 9.66e6·28-s + 3.41e6·29-s + 3.45e6·31-s − 2.88e7·32-s − 1.97e7·34-s + 6.92e6·35-s − 3.94e6·37-s + 2.16e7·38-s − 4.45e7·40-s + 1.76e7·41-s + ⋯ |
| L(s) = 1 | − 1.96·2-s + 2.86·4-s + 0.751·5-s + 1.03·7-s − 3.66·8-s − 1.47·10-s − 0.297·11-s + 0.0485·13-s − 2.03·14-s + 4.33·16-s + 1.29·17-s − 0.858·19-s + 2.15·20-s + 0.585·22-s + 0.0501·23-s − 0.434·25-s − 0.0955·26-s + 2.97·28-s + 0.897·29-s + 0.672·31-s − 4.86·32-s − 2.53·34-s + 0.780·35-s − 0.346·37-s + 1.68·38-s − 2.75·40-s + 0.973·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.9682770128\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9682770128\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + 44.4T + 512T^{2} \) |
| 5 | \( 1 - 1.05e3T + 1.95e6T^{2} \) |
| 7 | \( 1 - 6.59e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 1.44e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 5.00e3T + 1.06e10T^{2} \) |
| 17 | \( 1 - 4.44e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 4.87e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 6.73e4T + 1.80e12T^{2} \) |
| 29 | \( 1 - 3.41e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 3.45e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 3.94e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 1.76e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 3.30e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 3.71e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 7.86e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 5.13e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 1.92e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 2.14e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 3.25e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 1.57e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 3.58e7T + 1.19e17T^{2} \) |
| 83 | \( 1 + 8.07e6T + 1.86e17T^{2} \) |
| 89 | \( 1 + 9.43e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 9.67e8T + 7.60e17T^{2} \) |
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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
−15.69631300266965965298704642435, −14.39661355569706071050628325184, −12.15076212432037677024552483734, −10.84962267442712759189532283193, −9.925071924354711601506677777377, −8.610316448326135491177583777183, −7.55574876224531037726053753667, −5.91300102340919663309451511663, −2.33574633783176467565860962157, −1.03596924526436548316924677411,
1.03596924526436548316924677411, 2.33574633783176467565860962157, 5.91300102340919663309451511663, 7.55574876224531037726053753667, 8.610316448326135491177583777183, 9.925071924354711601506677777377, 10.84962267442712759189532283193, 12.15076212432037677024552483734, 14.39661355569706071050628325184, 15.69631300266965965298704642435
