| L(s) = 1 | − 3·3-s + 21.8·7-s + 9·9-s − 54.6·11-s − 82.7·13-s + 100.·17-s + 84.1·19-s − 65.6·21-s − 0.880·23-s − 27·27-s − 99.1·29-s + 78.9·31-s + 163.·33-s − 390.·37-s + 248.·39-s + 104.·41-s + 241.·43-s + 512.·47-s + 135.·49-s − 301.·51-s − 284.·53-s − 252.·57-s + 709.·59-s + 470.·61-s + 196.·63-s − 667.·67-s + 2.64·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.18·7-s + 0.333·9-s − 1.49·11-s − 1.76·13-s + 1.43·17-s + 1.01·19-s − 0.682·21-s − 0.00798·23-s − 0.192·27-s − 0.634·29-s + 0.457·31-s + 0.864·33-s − 1.73·37-s + 1.01·39-s + 0.398·41-s + 0.856·43-s + 1.59·47-s + 0.395·49-s − 0.827·51-s − 0.737·53-s − 0.586·57-s + 1.56·59-s + 0.987·61-s + 0.393·63-s − 1.21·67-s + 0.00460·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.539578577\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.539578577\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 3T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 21.8T + 343T^{2} \) |
| 11 | \( 1 + 54.6T + 1.33e3T^{2} \) |
| 13 | \( 1 + 82.7T + 2.19e3T^{2} \) |
| 17 | \( 1 - 100.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 84.1T + 6.85e3T^{2} \) |
| 23 | \( 1 + 0.880T + 1.21e4T^{2} \) |
| 29 | \( 1 + 99.1T + 2.43e4T^{2} \) |
| 31 | \( 1 - 78.9T + 2.97e4T^{2} \) |
| 37 | \( 1 + 390.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 104.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 241.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 512.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 284.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 709.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 470.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 667.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 51.5T + 3.57e5T^{2} \) |
| 73 | \( 1 + 371.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 79.3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 682.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 628.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.51e3T + 9.12e5T^{2} \) |
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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
−9.614861633699015811875276585000, −8.361007370706222854265598759609, −7.48033940212782732115677139539, −7.32201388233836877302955811624, −5.48153607394015608648403020307, −5.38067234569189429989976921286, −4.47379916314477792146923025764, −3.02118866492626896976584612935, −1.97322099357439992049498620944, −0.64920873248313295672391547249,
0.64920873248313295672391547249, 1.97322099357439992049498620944, 3.02118866492626896976584612935, 4.47379916314477792146923025764, 5.38067234569189429989976921286, 5.48153607394015608648403020307, 7.32201388233836877302955811624, 7.48033940212782732115677139539, 8.361007370706222854265598759609, 9.614861633699015811875276585000
