| L(s) = 1 | − 5-s − 11-s − 13-s + 6·17-s + 2·19-s + 7·23-s + 25-s + 5·29-s + 4·31-s − 10·37-s + 9·41-s − 3·43-s + 9·47-s − 9·53-s + 55-s + 12·59-s + 65-s − 8·67-s − 2·71-s + 2·73-s − 6·79-s + 4·83-s − 6·85-s + 6·89-s − 2·95-s + 16·97-s + 101-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.301·11-s − 0.277·13-s + 1.45·17-s + 0.458·19-s + 1.45·23-s + 1/5·25-s + 0.928·29-s + 0.718·31-s − 1.64·37-s + 1.40·41-s − 0.457·43-s + 1.31·47-s − 1.23·53-s + 0.134·55-s + 1.56·59-s + 0.124·65-s − 0.977·67-s − 0.237·71-s + 0.234·73-s − 0.675·79-s + 0.439·83-s − 0.650·85-s + 0.635·89-s − 0.205·95-s + 1.62·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 388080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 388080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.110887991\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.110887991\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 7 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 + 3 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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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
−12.48921356007367, −11.94585606494220, −11.71365579803730, −11.09965012257168, −10.61270587209145, −10.21244628549081, −9.896944348134082, −9.138558808469929, −8.946052004320081, −8.298879528226901, −7.858987222699287, −7.447537941056644, −7.043317777836364, −6.546319244728688, −5.904095023044927, −5.418924015523943, −4.980053893738374, −4.576335631974330, −3.889799053067529, −3.327276456071384, −2.958757387285842, −2.458234455034794, −1.599971500473394, −0.9735135649840547, −0.5452371722804592,
0.5452371722804592, 0.9735135649840547, 1.599971500473394, 2.458234455034794, 2.958757387285842, 3.327276456071384, 3.889799053067529, 4.576335631974330, 4.980053893738374, 5.418924015523943, 5.904095023044927, 6.546319244728688, 7.043317777836364, 7.447537941056644, 7.858987222699287, 8.298879528226901, 8.946052004320081, 9.138558808469929, 9.896944348134082, 10.21244628549081, 10.61270587209145, 11.09965012257168, 11.71365579803730, 11.94585606494220, 12.48921356007367
