| L(s) = 1 | − 1.32·2-s − 6.23·4-s − 15.4·5-s − 7.96·7-s + 18.9·8-s + 20.4·10-s − 12.7·11-s + 10.5·14-s + 24.8·16-s + 54·17-s − 84.5·19-s + 96.2·20-s + 16.9·22-s − 122.·23-s + 112.·25-s + 49.6·28-s − 140.·29-s − 116.·31-s − 184.·32-s − 71.6·34-s + 122.·35-s − 433.·37-s + 112.·38-s − 291.·40-s + 205.·41-s − 418.·43-s + 79.6·44-s + ⋯ |
| L(s) = 1 | − 0.469·2-s − 0.779·4-s − 1.37·5-s − 0.430·7-s + 0.835·8-s + 0.647·10-s − 0.350·11-s + 0.201·14-s + 0.387·16-s + 0.770·17-s − 1.02·19-s + 1.07·20-s + 0.164·22-s − 1.11·23-s + 0.903·25-s + 0.335·28-s − 0.901·29-s − 0.674·31-s − 1.01·32-s − 0.361·34-s + 0.593·35-s − 1.92·37-s + 0.479·38-s − 1.15·40-s + 0.784·41-s − 1.48·43-s + 0.272·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.04778467636\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.04778467636\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 1.32T + 8T^{2} \) |
| 5 | \( 1 + 15.4T + 125T^{2} \) |
| 7 | \( 1 + 7.96T + 343T^{2} \) |
| 11 | \( 1 + 12.7T + 1.33e3T^{2} \) |
| 17 | \( 1 - 54T + 4.91e3T^{2} \) |
| 19 | \( 1 + 84.5T + 6.85e3T^{2} \) |
| 23 | \( 1 + 122.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 140.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 116.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 433.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 205.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 418.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 485.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 674.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 186.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 671.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 14.0T + 3.00e5T^{2} \) |
| 71 | \( 1 - 346.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 832.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 335.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 568.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 236.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.27e3T + 9.12e5T^{2} \) |
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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
−8.937576754128155798912807405821, −8.287405454861771304471143348148, −7.73186414722305637621799340154, −6.97114664631482552189802232342, −5.75838264734123836280183254451, −4.79053798433035189212045160954, −3.91861308170087015614107261167, −3.34162545487194498416355188288, −1.67920503356369532511487561673, −0.11453834143139711145686727803,
0.11453834143139711145686727803, 1.67920503356369532511487561673, 3.34162545487194498416355188288, 3.91861308170087015614107261167, 4.79053798433035189212045160954, 5.75838264734123836280183254451, 6.97114664631482552189802232342, 7.73186414722305637621799340154, 8.287405454861771304471143348148, 8.937576754128155798912807405821
