| L(s) = 1 | + 2·3-s + 7-s + 9-s + 4·11-s − 4·13-s − 2·17-s + 6·19-s + 2·21-s − 8·23-s − 5·25-s − 4·27-s + 2·29-s + 4·31-s + 8·33-s + 10·37-s − 8·39-s − 10·41-s − 4·43-s − 4·47-s + 49-s − 4·51-s − 2·53-s + 12·57-s − 10·59-s − 8·61-s + 63-s + 8·67-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.377·7-s + 1/3·9-s + 1.20·11-s − 1.10·13-s − 0.485·17-s + 1.37·19-s + 0.436·21-s − 1.66·23-s − 25-s − 0.769·27-s + 0.371·29-s + 0.718·31-s + 1.39·33-s + 1.64·37-s − 1.28·39-s − 1.56·41-s − 0.609·43-s − 0.583·47-s + 1/7·49-s − 0.560·51-s − 0.274·53-s + 1.58·57-s − 1.30·59-s − 1.02·61-s + 0.125·63-s + 0.977·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.717518751\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.717518751\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 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
−12.06805530931609310079119258368, −11.57716090340427312946484264281, −9.913173607034066853189935309323, −9.389652658008486485429618465354, −8.247827016681358338699106327510, −7.53632864653844024945923131798, −6.19321040018157050389604631782, −4.62285613804853643208241606946, −3.39651452170084757716747861999, −2.00516068697428410265218115685,
2.00516068697428410265218115685, 3.39651452170084757716747861999, 4.62285613804853643208241606946, 6.19321040018157050389604631782, 7.53632864653844024945923131798, 8.247827016681358338699106327510, 9.389652658008486485429618465354, 9.913173607034066853189935309323, 11.57716090340427312946484264281, 12.06805530931609310079119258368
