| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 7-s − 8-s + 9-s + 4·11-s + 12-s − 13-s + 14-s + 16-s + 17-s − 18-s − 8·19-s − 21-s − 4·22-s − 8·23-s − 24-s + 26-s + 27-s − 28-s + 10·29-s − 32-s + 4·33-s − 34-s + 36-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s + 1.20·11-s + 0.288·12-s − 0.277·13-s + 0.267·14-s + 1/4·16-s + 0.242·17-s − 0.235·18-s − 1.83·19-s − 0.218·21-s − 0.852·22-s − 1.66·23-s − 0.204·24-s + 0.196·26-s + 0.192·27-s − 0.188·28-s + 1.85·29-s − 0.176·32-s + 0.696·33-s − 0.171·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 232050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 232050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 11 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−12.97982840060254, −12.73899852335185, −12.11965736964478, −11.81159139546891, −11.31968372269433, −10.62365825701667, −10.24729808453068, −9.890872686978359, −9.325948856737764, −9.028992355457339, −8.402422104466634, −8.087039847733934, −7.702965418323037, −6.891974773634842, −6.577986037816194, −6.126520443739190, −5.790309214360887, −4.567252196038217, −4.422109254723799, −3.881663416367338, −3.085493903683687, −2.727275542909220, −1.967925800643941, −1.582437788526761, −0.7758474443153678, 0,
0.7758474443153678, 1.582437788526761, 1.967925800643941, 2.727275542909220, 3.085493903683687, 3.881663416367338, 4.422109254723799, 4.567252196038217, 5.790309214360887, 6.126520443739190, 6.577986037816194, 6.891974773634842, 7.702965418323037, 8.087039847733934, 8.402422104466634, 9.028992355457339, 9.325948856737764, 9.890872686978359, 10.24729808453068, 10.62365825701667, 11.31968372269433, 11.81159139546891, 12.11965736964478, 12.73899852335185, 12.97982840060254
