| L(s) = 1 | − 5-s + 3.64·7-s − 11-s + 3.64·13-s + 1.64·17-s + 2·19-s + 25-s − 3.29·29-s − 1.29·31-s − 3.64·35-s − 1.29·37-s − 3.29·41-s + 0.354·43-s + 3.29·47-s + 6.29·49-s + 12.5·53-s + 55-s + 12.5·59-s + 5.29·61-s − 3.64·65-s + 8·67-s − 12.5·71-s − 2.93·73-s − 3.64·77-s − 4.58·79-s − 4.93·83-s − 1.64·85-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.37·7-s − 0.301·11-s + 1.01·13-s + 0.399·17-s + 0.458·19-s + 0.200·25-s − 0.611·29-s − 0.231·31-s − 0.616·35-s − 0.212·37-s − 0.514·41-s + 0.0540·43-s + 0.480·47-s + 0.898·49-s + 1.72·53-s + 0.134·55-s + 1.63·59-s + 0.677·61-s − 0.452·65-s + 0.977·67-s − 1.49·71-s − 0.343·73-s − 0.415·77-s − 0.515·79-s − 0.541·83-s − 0.178·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.005831331\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.005831331\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| good | 7 | \( 1 - 3.64T + 7T^{2} \) |
| 13 | \( 1 - 3.64T + 13T^{2} \) |
| 17 | \( 1 - 1.64T + 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 3.29T + 29T^{2} \) |
| 31 | \( 1 + 1.29T + 31T^{2} \) |
| 37 | \( 1 + 1.29T + 37T^{2} \) |
| 41 | \( 1 + 3.29T + 41T^{2} \) |
| 43 | \( 1 - 0.354T + 43T^{2} \) |
| 47 | \( 1 - 3.29T + 47T^{2} \) |
| 53 | \( 1 - 12.5T + 53T^{2} \) |
| 59 | \( 1 - 12.5T + 59T^{2} \) |
| 61 | \( 1 - 5.29T + 61T^{2} \) |
| 67 | \( 1 - 8T + 67T^{2} \) |
| 71 | \( 1 + 12.5T + 71T^{2} \) |
| 73 | \( 1 + 2.93T + 73T^{2} \) |
| 79 | \( 1 + 4.58T + 79T^{2} \) |
| 83 | \( 1 + 4.93T + 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 - 8.58T + 97T^{2} \) |
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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
−8.883597157683855596895771824359, −8.414208185134297247405459564916, −7.66653544780061977518956140324, −7.02574455051330314678956452630, −5.79430249420584356701857850004, −5.18605571380262338551642717283, −4.23136545445714534525683944742, −3.42916454373224746581135804257, −2.10907696129510740237839293793, −1.01152591419051378837832409053,
1.01152591419051378837832409053, 2.10907696129510740237839293793, 3.42916454373224746581135804257, 4.23136545445714534525683944742, 5.18605571380262338551642717283, 5.79430249420584356701857850004, 7.02574455051330314678956452630, 7.66653544780061977518956140324, 8.414208185134297247405459564916, 8.883597157683855596895771824359
