| L(s) = 1 | + 2·2-s + 2·4-s − 5-s − 2·10-s + 4·11-s + 2·13-s − 4·16-s + 3·17-s + 8·19-s − 2·20-s + 8·22-s + 6·23-s − 4·25-s + 4·26-s + 4·29-s − 6·31-s − 8·32-s + 6·34-s − 3·37-s + 16·38-s + 41-s + 11·43-s + 8·44-s + 12·46-s + 9·47-s − 8·50-s + 4·52-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s − 0.447·5-s − 0.632·10-s + 1.20·11-s + 0.554·13-s − 16-s + 0.727·17-s + 1.83·19-s − 0.447·20-s + 1.70·22-s + 1.25·23-s − 4/5·25-s + 0.784·26-s + 0.742·29-s − 1.07·31-s − 1.41·32-s + 1.02·34-s − 0.493·37-s + 2.59·38-s + 0.156·41-s + 1.67·43-s + 1.20·44-s + 1.76·46-s + 1.31·47-s − 1.13·50-s + 0.554·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.539577585\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.539577585\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - p T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 - T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 15 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−9.425361220593281165848466844703, −9.022526521140030205636098726414, −7.70452327645607948018799635565, −7.02739091275998416273848969818, −6.03726056796490680523038320077, −5.40364055003978758098114460014, −4.40677406563599377748102980133, −3.62857038648496213368827109078, −2.97546835317207025312612260349, −1.26108523165476729822161219566,
1.26108523165476729822161219566, 2.97546835317207025312612260349, 3.62857038648496213368827109078, 4.40677406563599377748102980133, 5.40364055003978758098114460014, 6.03726056796490680523038320077, 7.02739091275998416273848969818, 7.70452327645607948018799635565, 9.022526521140030205636098726414, 9.425361220593281165848466844703
