| L(s) = 1 | − 3-s + 9-s − 4·11-s + 2·13-s − 6·17-s − 8·23-s − 27-s − 10·29-s − 8·31-s + 4·33-s + 2·37-s − 2·39-s + 2·41-s − 8·43-s − 4·47-s + 6·51-s + 10·53-s − 4·59-s − 6·61-s + 8·69-s + 12·71-s − 6·73-s + 8·79-s + 81-s − 4·83-s + 10·87-s − 14·89-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s − 1.20·11-s + 0.554·13-s − 1.45·17-s − 1.66·23-s − 0.192·27-s − 1.85·29-s − 1.43·31-s + 0.696·33-s + 0.328·37-s − 0.320·39-s + 0.312·41-s − 1.21·43-s − 0.583·47-s + 0.840·51-s + 1.37·53-s − 0.520·59-s − 0.768·61-s + 0.963·69-s + 1.42·71-s − 0.702·73-s + 0.900·79-s + 1/9·81-s − 0.439·83-s + 1.07·87-s − 1.48·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−13.05300783470873, −12.81400344153077, −12.23491241538450, −11.61968398903271, −11.25900277306756, −10.82026643470849, −10.57370081732989, −9.769376687148976, −9.645606491004036, −8.820600894982776, −8.522206559905722, −7.871631689255551, −7.468652455271333, −7.007119916543630, −6.372989642236965, −5.886017207527925, −5.551469230438636, −4.981543536641800, −4.424984965405175, −3.860473313242320, −3.451370808031402, −2.589534328337348, −1.939159701114485, −1.722066939527156, −0.4991754004227342, 0,
0.4991754004227342, 1.722066939527156, 1.939159701114485, 2.589534328337348, 3.451370808031402, 3.860473313242320, 4.424984965405175, 4.981543536641800, 5.551469230438636, 5.886017207527925, 6.372989642236965, 7.007119916543630, 7.468652455271333, 7.871631689255551, 8.522206559905722, 8.820600894982776, 9.645606491004036, 9.769376687148976, 10.57370081732989, 10.82026643470849, 11.25900277306756, 11.61968398903271, 12.23491241538450, 12.81400344153077, 13.05300783470873
