L(s) = 1 | − 3·3-s − 2·7-s + 6·9-s − 11-s + 5·17-s − 19-s + 6·21-s − 2·23-s − 9·27-s − 8·29-s − 10·31-s + 3·33-s + 6·37-s + 3·41-s + 4·43-s − 4·47-s − 3·49-s − 15·51-s + 6·53-s + 3·57-s − 8·59-s + 10·61-s − 12·63-s + 67-s + 6·69-s + 12·71-s − 3·73-s + ⋯ |
L(s) = 1 | − 1.73·3-s − 0.755·7-s + 2·9-s − 0.301·11-s + 1.21·17-s − 0.229·19-s + 1.30·21-s − 0.417·23-s − 1.73·27-s − 1.48·29-s − 1.79·31-s + 0.522·33-s + 0.986·37-s + 0.468·41-s + 0.609·43-s − 0.583·47-s − 3/7·49-s − 2.10·51-s + 0.824·53-s + 0.397·57-s − 1.04·59-s + 1.28·61-s − 1.51·63-s + 0.122·67-s + 0.722·69-s + 1.42·71-s − 0.351·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 \) |
good | 3 | \( 1 + p T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 3 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 - 13 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 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
−15.41855267672656, −14.71643717923819, −14.36695000721500, −13.34329829925200, −12.95723881991262, −12.63720203294584, −12.03976051846063, −11.58807480815900, −10.93660004543650, −10.67365022396794, −9.997598506291374, −9.515387294301773, −9.080340850951967, −7.888166268268016, −7.609544977660288, −6.916646806254240, −6.287645685303578, −5.890076082734625, −5.321958417395171, −4.939202736483904, −3.847370322656267, −3.672337025044855, −2.503825911166073, −1.626982137470928, −0.7368125701643577, 0,
0.7368125701643577, 1.626982137470928, 2.503825911166073, 3.672337025044855, 3.847370322656267, 4.939202736483904, 5.321958417395171, 5.890076082734625, 6.287645685303578, 6.916646806254240, 7.609544977660288, 7.888166268268016, 9.080340850951967, 9.515387294301773, 9.997598506291374, 10.67365022396794, 10.93660004543650, 11.58807480815900, 12.03976051846063, 12.63720203294584, 12.95723881991262, 13.34329829925200, 14.36695000721500, 14.71643717923819, 15.41855267672656