L(s) = 1 | − 2-s + 4-s − 7-s − 8-s − 2·13-s + 14-s + 16-s − 17-s + 4·19-s + 6·23-s + 2·26-s − 28-s − 2·29-s + 6·31-s − 32-s + 34-s + 6·37-s − 4·38-s + 2·41-s + 10·43-s − 6·46-s + 12·47-s + 49-s − 2·52-s − 6·53-s + 56-s + 2·58-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.377·7-s − 0.353·8-s − 0.554·13-s + 0.267·14-s + 1/4·16-s − 0.242·17-s + 0.917·19-s + 1.25·23-s + 0.392·26-s − 0.188·28-s − 0.371·29-s + 1.07·31-s − 0.176·32-s + 0.171·34-s + 0.986·37-s − 0.648·38-s + 0.312·41-s + 1.52·43-s − 0.884·46-s + 1.75·47-s + 1/7·49-s − 0.277·52-s − 0.824·53-s + 0.133·56-s + 0.262·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 53550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 53550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.920822604\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.920822604\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 + T \) |
good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 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
−14.52645509524123, −13.97641330734116, −13.38837838709586, −12.81595643481180, −12.41241358625498, −11.80382573787244, −11.25723433968317, −10.88756886351477, −10.21499873627762, −9.692337108275937, −9.326964528834781, −8.840456200160873, −8.143507240773766, −7.656900479336189, −7.030759022585730, −6.768604527097398, −5.872384291027813, −5.507242169606889, −4.718089483757937, −4.091023776169689, −3.304743613590390, −2.644492011444004, −2.221476982246649, −1.041702730038711, −0.6589747759448386,
0.6589747759448386, 1.041702730038711, 2.221476982246649, 2.644492011444004, 3.304743613590390, 4.091023776169689, 4.718089483757937, 5.507242169606889, 5.872384291027813, 6.768604527097398, 7.030759022585730, 7.656900479336189, 8.143507240773766, 8.840456200160873, 9.326964528834781, 9.692337108275937, 10.21499873627762, 10.88756886351477, 11.25723433968317, 11.80382573787244, 12.41241358625498, 12.81595643481180, 13.38837838709586, 13.97641330734116, 14.52645509524123