L(s) = 1 | + 20.7·2-s − 29.4·3-s + 303.·4-s + 223.·5-s − 612.·6-s − 1.41e3·7-s + 3.64e3·8-s − 1.31e3·9-s + 4.64e3·10-s − 1.33e3·11-s − 8.94e3·12-s + 975.·13-s − 2.92e4·14-s − 6.59e3·15-s + 3.67e4·16-s + 3.67e4·17-s − 2.73e4·18-s − 2.06e4·19-s + 6.77e4·20-s + 4.15e4·21-s − 2.76e4·22-s + 3.49e4·23-s − 1.07e5·24-s − 2.81e4·25-s + 2.02e4·26-s + 1.03e5·27-s − 4.27e5·28-s + ⋯ |
L(s) = 1 | + 1.83·2-s − 0.630·3-s + 2.36·4-s + 0.799·5-s − 1.15·6-s − 1.55·7-s + 2.51·8-s − 0.602·9-s + 1.46·10-s − 0.301·11-s − 1.49·12-s + 0.123·13-s − 2.85·14-s − 0.504·15-s + 2.24·16-s + 1.81·17-s − 1.10·18-s − 0.690·19-s + 1.89·20-s + 0.980·21-s − 0.553·22-s + 0.598·23-s − 1.58·24-s − 0.360·25-s + 0.226·26-s + 1.01·27-s − 3.68·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(4)\) |
\(\approx\) |
\(2.993756932\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.993756932\) |
\(L(\frac{9}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 + 1.33e3T \) |
good | 2 | \( 1 - 20.7T + 128T^{2} \) |
| 3 | \( 1 + 29.4T + 2.18e3T^{2} \) |
| 5 | \( 1 - 223.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 1.41e3T + 8.23e5T^{2} \) |
| 13 | \( 1 - 975.T + 6.27e7T^{2} \) |
| 17 | \( 1 - 3.67e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 2.06e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 3.49e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.21e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 5.99e4T + 2.75e10T^{2} \) |
| 37 | \( 1 + 3.65e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 1.68e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 4.23e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 4.35e4T + 5.06e11T^{2} \) |
| 53 | \( 1 + 8.99e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 8.53e4T + 2.48e12T^{2} \) |
| 61 | \( 1 - 1.63e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 6.85e5T + 6.06e12T^{2} \) |
| 71 | \( 1 + 3.52e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 9.00e5T + 1.10e13T^{2} \) |
| 79 | \( 1 - 1.56e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 6.62e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 5.04e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.22e7T + 8.07e13T^{2} \) |
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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
−19.31875084418176305168685974675, −16.98447667395267982327100981369, −15.91952827231865607266380639402, −14.28500223929598701703553480126, −13.10122524915071680121589525414, −12.06560535172230628117222964736, −10.27986800613411480287336390760, −6.47457005282767244384449341095, −5.50343168201262409154254978575, −3.09199739069876330913918495664,
3.09199739069876330913918495664, 5.50343168201262409154254978575, 6.47457005282767244384449341095, 10.27986800613411480287336390760, 12.06560535172230628117222964736, 13.10122524915071680121589525414, 14.28500223929598701703553480126, 15.91952827231865607266380639402, 16.98447667395267982327100981369, 19.31875084418176305168685974675