L(s) = 1 | + 17.3·2-s + 73.8·3-s + 172.·4-s − 122.·5-s + 1.28e3·6-s + 247.·7-s + 777.·8-s + 3.26e3·9-s − 2.13e3·10-s + 2.50e3·11-s + 1.27e4·12-s + 1.37e3·13-s + 4.28e3·14-s − 9.07e3·15-s − 8.63e3·16-s − 3.20e4·17-s + 5.66e4·18-s + 1.02e4·19-s − 2.12e4·20-s + 1.82e4·21-s + 4.34e4·22-s − 2.18e4·23-s + 5.73e4·24-s − 6.30e4·25-s + 2.38e4·26-s + 7.95e4·27-s + 4.26e4·28-s + ⋯ |
L(s) = 1 | + 1.53·2-s + 1.57·3-s + 1.35·4-s − 0.439·5-s + 2.42·6-s + 0.272·7-s + 0.536·8-s + 1.49·9-s − 0.674·10-s + 0.567·11-s + 2.13·12-s + 0.173·13-s + 0.417·14-s − 0.694·15-s − 0.527·16-s − 1.58·17-s + 2.28·18-s + 0.342·19-s − 0.593·20-s + 0.429·21-s + 0.869·22-s − 0.374·23-s + 0.847·24-s − 0.806·25-s + 0.266·26-s + 0.778·27-s + 0.367·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(4)\) |
\(\approx\) |
\(6.016787041\) |
\(L(\frac12)\) |
\(\approx\) |
\(6.016787041\) |
\(L(\frac{9}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 43 | \( 1 + 7.95e4T \) |
good | 2 | \( 1 - 17.3T + 128T^{2} \) |
| 3 | \( 1 - 73.8T + 2.18e3T^{2} \) |
| 5 | \( 1 + 122.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 247.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 2.50e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 1.37e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 3.20e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 1.02e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 2.18e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.73e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.60e4T + 2.75e10T^{2} \) |
| 37 | \( 1 + 7.56e4T + 9.49e10T^{2} \) |
| 41 | \( 1 - 4.52e5T + 1.94e11T^{2} \) |
| 47 | \( 1 - 1.13e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.42e6T + 1.17e12T^{2} \) |
| 59 | \( 1 - 1.48e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 2.69e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 2.19e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 4.77e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 4.98e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 3.39e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 8.35e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 7.83e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 6.12e6T + 8.07e13T^{2} \) |
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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
−14.17537952117450683505998029843, −13.68046648085650842306426212162, −12.51933690462303138029463699215, −11.27557701141838662590613316987, −9.295012032737493275869982145081, −8.079134091288438434666495034462, −6.57880897046850835067299464098, −4.53476119191405497457225835453, −3.54342324895824396165458645725, −2.22070434964846614755690740934,
2.22070434964846614755690740934, 3.54342324895824396165458645725, 4.53476119191405497457225835453, 6.57880897046850835067299464098, 8.079134091288438434666495034462, 9.295012032737493275869982145081, 11.27557701141838662590613316987, 12.51933690462303138029463699215, 13.68046648085650842306426212162, 14.17537952117450683505998029843