| L(s) = 1 | − 2·4-s + 5-s − 7-s + 2·13-s + 4·16-s − 6·17-s − 7·19-s − 2·20-s + 6·23-s + 25-s + 2·28-s − 31-s − 35-s − 7·37-s + 6·41-s + 8·43-s − 6·49-s − 4·52-s + 6·53-s + 12·59-s − 61-s − 8·64-s + 2·65-s − 7·67-s + 12·68-s − 6·71-s − 13·73-s + ⋯ |
| L(s) = 1 | − 4-s + 0.447·5-s − 0.377·7-s + 0.554·13-s + 16-s − 1.45·17-s − 1.60·19-s − 0.447·20-s + 1.25·23-s + 1/5·25-s + 0.377·28-s − 0.179·31-s − 0.169·35-s − 1.15·37-s + 0.937·41-s + 1.21·43-s − 6/7·49-s − 0.554·52-s + 0.824·53-s + 1.56·59-s − 0.128·61-s − 64-s + 0.248·65-s − 0.855·67-s + 1.45·68-s − 0.712·71-s − 1.52·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5445 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5445 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.200307971\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.200307971\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 13 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−8.485453916851019492858569664820, −7.43711424473686269431816778687, −6.61280187235421899316911179550, −6.05793013905856029986241325619, −5.20779791692855593559951650094, −4.44793309957623836809852653633, −3.87307626331343778310808010397, −2.84635947998777897362121685781, −1.86441514423774169059856966061, −0.58620388806846202709441772114,
0.58620388806846202709441772114, 1.86441514423774169059856966061, 2.84635947998777897362121685781, 3.87307626331343778310808010397, 4.44793309957623836809852653633, 5.20779791692855593559951650094, 6.05793013905856029986241325619, 6.61280187235421899316911179550, 7.43711424473686269431816778687, 8.485453916851019492858569664820
