L(s) = 1 | − 2·5-s − 2·7-s − 11-s + 2·13-s − 4·17-s − 25-s − 8·29-s + 8·31-s + 4·35-s − 10·37-s + 8·41-s − 2·43-s + 8·47-s − 3·49-s − 2·53-s + 2·55-s + 12·59-s + 10·61-s − 4·65-s − 12·67-s + 8·71-s + 6·73-s + 2·77-s + 2·79-s − 16·83-s + 8·85-s − 14·89-s + ⋯ |
L(s) = 1 | − 0.894·5-s − 0.755·7-s − 0.301·11-s + 0.554·13-s − 0.970·17-s − 1/5·25-s − 1.48·29-s + 1.43·31-s + 0.676·35-s − 1.64·37-s + 1.24·41-s − 0.304·43-s + 1.16·47-s − 3/7·49-s − 0.274·53-s + 0.269·55-s + 1.56·59-s + 1.28·61-s − 0.496·65-s − 1.46·67-s + 0.949·71-s + 0.702·73-s + 0.227·77-s + 0.225·79-s − 1.75·83-s + 0.867·85-s − 1.48·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 142956 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 142956 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7036787840\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7036787840\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 \) |
good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 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
−13.27998517504297, −12.86138835182869, −12.57878874273790, −11.84888084140897, −11.48600058283197, −11.10451176946132, −10.53888373765424, −10.03802380331652, −9.547517827010959, −8.959054631179533, −8.508163893829714, −8.114219100166644, −7.435899042731261, −7.035572495028349, −6.584693094695073, −5.899816206018027, −5.542730508121112, −4.732002830564585, −4.228977671974347, −3.697407272771279, −3.322352965058699, −2.529814254886539, −2.018760649207711, −1.061817681511914, −0.2766228040293899,
0.2766228040293899, 1.061817681511914, 2.018760649207711, 2.529814254886539, 3.322352965058699, 3.697407272771279, 4.228977671974347, 4.732002830564585, 5.542730508121112, 5.899816206018027, 6.584693094695073, 7.035572495028349, 7.435899042731261, 8.114219100166644, 8.508163893829714, 8.959054631179533, 9.547517827010959, 10.03802380331652, 10.53888373765424, 11.10451176946132, 11.48600058283197, 11.84888084140897, 12.57878874273790, 12.86138835182869, 13.27998517504297