| L(s) = 1 | + 2-s + 2·3-s + 4-s + 5-s + 2·6-s + 8-s + 9-s + 10-s + 3·11-s + 2·12-s − 5·13-s + 2·15-s + 16-s − 6·17-s + 18-s + 19-s + 20-s + 3·22-s + 3·23-s + 2·24-s + 25-s − 5·26-s − 4·27-s − 6·29-s + 2·30-s + 4·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.15·3-s + 1/2·4-s + 0.447·5-s + 0.816·6-s + 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.904·11-s + 0.577·12-s − 1.38·13-s + 0.516·15-s + 1/4·16-s − 1.45·17-s + 0.235·18-s + 0.229·19-s + 0.223·20-s + 0.639·22-s + 0.625·23-s + 0.408·24-s + 1/5·25-s − 0.980·26-s − 0.769·27-s − 1.11·29-s + 0.365·30-s + 0.718·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.092793554\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.092793554\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 11 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−11.14507794859466678682828912363, −9.808705402590549548523141273172, −9.263851768447220146642287460280, −8.289976858527741605754067242124, −7.22555857938969291949122618245, −6.41172881162855055433374816405, −5.10121929575789116373472283151, −4.09357447562145058841583904603, −2.89619370582642367784100306287, −2.01395492425859231821166088910,
2.01395492425859231821166088910, 2.89619370582642367784100306287, 4.09357447562145058841583904603, 5.10121929575789116373472283151, 6.41172881162855055433374816405, 7.22555857938969291949122618245, 8.289976858527741605754067242124, 9.263851768447220146642287460280, 9.808705402590549548523141273172, 11.14507794859466678682828912363
