| L(s) = 1 | + 31.8·2-s − 261.·3-s + 501.·4-s − 1.90e3·5-s − 8.33e3·6-s + 3.06e3·7-s − 319.·8-s + 4.88e4·9-s − 6.07e4·10-s − 1.46e4·11-s − 1.31e5·12-s − 1.00e5·13-s + 9.74e4·14-s + 4.99e5·15-s − 2.67e5·16-s + 1.41e5·17-s + 1.55e6·18-s − 4.91e5·19-s − 9.57e5·20-s − 8.01e5·21-s − 4.66e5·22-s + 1.02e6·23-s + 8.36e4·24-s + 1.68e6·25-s − 3.21e6·26-s − 7.65e6·27-s + 1.53e6·28-s + ⋯ |
| L(s) = 1 | + 1.40·2-s − 1.86·3-s + 0.980·4-s − 1.36·5-s − 2.62·6-s + 0.481·7-s − 0.0275·8-s + 2.48·9-s − 1.92·10-s − 0.301·11-s − 1.83·12-s − 0.979·13-s + 0.677·14-s + 2.54·15-s − 1.01·16-s + 0.410·17-s + 3.49·18-s − 0.864·19-s − 1.33·20-s − 0.899·21-s − 0.424·22-s + 0.760·23-s + 0.0514·24-s + 0.864·25-s − 1.37·26-s − 2.77·27-s + 0.472·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + 1.46e4T \) |
| good | 2 | \( 1 - 31.8T + 512T^{2} \) |
| 3 | \( 1 + 261.T + 1.96e4T^{2} \) |
| 5 | \( 1 + 1.90e3T + 1.95e6T^{2} \) |
| 7 | \( 1 - 3.06e3T + 4.03e7T^{2} \) |
| 13 | \( 1 + 1.00e5T + 1.06e10T^{2} \) |
| 17 | \( 1 - 1.41e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 4.91e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 1.02e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 1.58e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 2.39e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.40e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + 8.90e6T + 3.27e14T^{2} \) |
| 43 | \( 1 - 3.79e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 3.02e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 1.24e6T + 3.29e15T^{2} \) |
| 59 | \( 1 - 1.27e8T + 8.66e15T^{2} \) |
| 61 | \( 1 + 6.35e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 7.51e7T + 2.72e16T^{2} \) |
| 71 | \( 1 + 9.00e7T + 4.58e16T^{2} \) |
| 73 | \( 1 + 1.51e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 5.42e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 4.30e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 5.75e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 2.55e8T + 7.60e17T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−17.36006124867685946634302307651, −15.99442768024064753403546243900, −14.93470858903509320241021772110, −12.68649601141333211969609081861, −11.92342239150368078271068301193, −10.91930781962024607604700133447, −7.13978149903749351023969810175, −5.35909162917206773108418184488, −4.25399431175674183303402831501, 0,
4.25399431175674183303402831501, 5.35909162917206773108418184488, 7.13978149903749351023969810175, 10.91930781962024607604700133447, 11.92342239150368078271068301193, 12.68649601141333211969609081861, 14.93470858903509320241021772110, 15.99442768024064753403546243900, 17.36006124867685946634302307651
