| 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
| \(L(2)\) |
\(\approx\) |
\(3.284769913\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.284769913\) |
| \(L(\frac{5}{2})\) |
|
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
−9.307180818851219892079030472775, −8.388918459351090839192016847280, −7.55751131997804371363681762160, −6.29721064039630347016816338798, −5.70776736150244280323402271041, −5.08633098353777881636386391772, −4.11462068124171021169062391522, −3.10532885501930015910677477839, −1.93389277118341985675288936185, −0.864712926976473255208328020719,
0.864712926976473255208328020719, 1.93389277118341985675288936185, 3.10532885501930015910677477839, 4.11462068124171021169062391522, 5.08633098353777881636386391772, 5.70776736150244280323402271041, 6.29721064039630347016816338798, 7.55751131997804371363681762160, 8.388918459351090839192016847280, 9.307180818851219892079030472775
