L(s) = 1 | + 2-s − 3-s + 4-s − 6-s − 2·7-s + 8-s − 2·9-s − 3·11-s − 12-s + 4·13-s − 2·14-s + 16-s + 3·17-s − 2·18-s + 5·19-s + 2·21-s − 3·22-s − 6·23-s − 24-s + 4·26-s + 5·27-s − 2·28-s + 2·31-s + 32-s + 3·33-s + 3·34-s − 2·36-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.755·7-s + 0.353·8-s − 2/3·9-s − 0.904·11-s − 0.288·12-s + 1.10·13-s − 0.534·14-s + 1/4·16-s + 0.727·17-s − 0.471·18-s + 1.14·19-s + 0.436·21-s − 0.639·22-s − 1.25·23-s − 0.204·24-s + 0.784·26-s + 0.962·27-s − 0.377·28-s + 0.359·31-s + 0.176·32-s + 0.522·33-s + 0.514·34-s − 1/3·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9568112265\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9568112265\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
good | 3 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 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
−15.85587358322534319659703271336, −14.27543395311716649730892535101, −13.31818220322005518851751496924, −12.18462032318833367073561378768, −11.16007422368021302024475397710, −9.930723730317650982781533521356, −8.071729218562685355881418106440, −6.35306147256574072432179047412, −5.34945783064988058695168525193, −3.29699975126484092788987476482,
3.29699975126484092788987476482, 5.34945783064988058695168525193, 6.35306147256574072432179047412, 8.071729218562685355881418106440, 9.930723730317650982781533521356, 11.16007422368021302024475397710, 12.18462032318833367073561378768, 13.31818220322005518851751496924, 14.27543395311716649730892535101, 15.85587358322534319659703271336