| L(s) = 1 | − 9.87·2-s + 20.2·3-s + 65.4·4-s + 73.8·5-s − 199.·6-s − 245.·7-s − 330.·8-s + 167.·9-s − 729.·10-s + 298.·11-s + 1.32e3·12-s + 141.·13-s + 2.42e3·14-s + 1.49e3·15-s + 1.16e3·16-s − 2.16e3·17-s − 1.65e3·18-s − 806.·19-s + 4.83e3·20-s − 4.98e3·21-s − 2.94e3·22-s + 1.58e3·23-s − 6.69e3·24-s + 2.32e3·25-s − 1.39e3·26-s − 1.53e3·27-s − 1.61e4·28-s + ⋯ |
| L(s) = 1 | − 1.74·2-s + 1.29·3-s + 2.04·4-s + 1.32·5-s − 2.26·6-s − 1.89·7-s − 1.82·8-s + 0.688·9-s − 2.30·10-s + 0.743·11-s + 2.65·12-s + 0.231·13-s + 3.31·14-s + 1.71·15-s + 1.14·16-s − 1.82·17-s − 1.20·18-s − 0.512·19-s + 2.70·20-s − 2.46·21-s − 1.29·22-s + 0.623·23-s − 2.37·24-s + 0.745·25-s − 0.404·26-s − 0.404·27-s − 3.88·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.394256673\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.394256673\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 547 | \( 1 - 2.99e5T \) |
| good | 2 | \( 1 + 9.87T + 32T^{2} \) |
| 3 | \( 1 - 20.2T + 243T^{2} \) |
| 5 | \( 1 - 73.8T + 3.12e3T^{2} \) |
| 7 | \( 1 + 245.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 298.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 141.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 2.16e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 806.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 1.58e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 7.21e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 4.70e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 5.29e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.55e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 9.07e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.03e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.70e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.35e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.46e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 5.44e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 1.48e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 5.88e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 7.15e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 9.58e3T + 3.93e9T^{2} \) |
| 89 | \( 1 - 8.15e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 4.58e4T + 8.58e9T^{2} \) |
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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
−9.469410171979707364524524607492, −9.221192519338837769463513831749, −8.828723699154088750398563201238, −7.52181634428304295097638303875, −6.55016005576318796685999832956, −6.15151921637953949283051174751, −3.80756101873225993143543243815, −2.53209276696698478822757442354, −2.14653508309583459472542106587, −0.68257005813297931254026512948,
0.68257005813297931254026512948, 2.14653508309583459472542106587, 2.53209276696698478822757442354, 3.80756101873225993143543243815, 6.15151921637953949283051174751, 6.55016005576318796685999832956, 7.52181634428304295097638303875, 8.828723699154088750398563201238, 9.221192519338837769463513831749, 9.469410171979707364524524607492
