| L(s) = 1 | + 2.46·2-s − 1.93·4-s + 11.2·5-s + 18.6·7-s − 24.4·8-s + 27.7·10-s + 16.5·11-s + 24.7·13-s + 45.9·14-s − 44.7·16-s + 125.·17-s − 104.·19-s − 21.8·20-s + 40.6·22-s − 116.·23-s + 2.01·25-s + 60.8·26-s − 36.1·28-s + 295.·29-s − 62.8·31-s + 85.5·32-s + 308.·34-s + 210.·35-s + 318.·37-s − 258.·38-s − 275.·40-s + 488.·41-s + ⋯ |
| L(s) = 1 | + 0.870·2-s − 0.242·4-s + 1.00·5-s + 1.00·7-s − 1.08·8-s + 0.877·10-s + 0.452·11-s + 0.527·13-s + 0.877·14-s − 0.699·16-s + 1.78·17-s − 1.26·19-s − 0.244·20-s + 0.393·22-s − 1.05·23-s + 0.0161·25-s + 0.458·26-s − 0.243·28-s + 1.89·29-s − 0.364·31-s + 0.472·32-s + 1.55·34-s + 1.01·35-s + 1.41·37-s − 1.10·38-s − 1.09·40-s + 1.86·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.748482384\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.748482384\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 59 | \( 1 - 59T \) |
| good | 2 | \( 1 - 2.46T + 8T^{2} \) |
| 5 | \( 1 - 11.2T + 125T^{2} \) |
| 7 | \( 1 - 18.6T + 343T^{2} \) |
| 11 | \( 1 - 16.5T + 1.33e3T^{2} \) |
| 13 | \( 1 - 24.7T + 2.19e3T^{2} \) |
| 17 | \( 1 - 125.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 104.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 116.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 295.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 62.8T + 2.97e4T^{2} \) |
| 37 | \( 1 - 318.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 488.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 127.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 221.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 40.8T + 1.48e5T^{2} \) |
| 61 | \( 1 + 237.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 532.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 634.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 375.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 323.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.17e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.44e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.04e3T + 9.12e5T^{2} \) |
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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
−10.38544054680911585680640958989, −9.628007399633379306343958497949, −8.640587151424728176195687637671, −7.85229224519590761570574600550, −6.23008676901418228412707239988, −5.81261241932580377887795562255, −4.74296594694877253047023436331, −3.89065034552378371381464183674, −2.49316396858221413581337714504, −1.15576373542972204668527009372,
1.15576373542972204668527009372, 2.49316396858221413581337714504, 3.89065034552378371381464183674, 4.74296594694877253047023436331, 5.81261241932580377887795562255, 6.23008676901418228412707239988, 7.85229224519590761570574600550, 8.640587151424728176195687637671, 9.628007399633379306343958497949, 10.38544054680911585680640958989
