| L(s) = 1 | + 3-s + 2·7-s + 9-s + 2·11-s − 2·13-s + 6·17-s + 8·19-s + 2·21-s + 4·23-s + 27-s − 8·29-s + 2·33-s + 10·37-s − 2·39-s + 2·41-s − 12·43-s − 3·49-s + 6·51-s − 10·53-s + 8·57-s − 6·59-s − 2·61-s + 2·63-s − 8·67-s + 4·69-s + 4·71-s + 4·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.755·7-s + 1/3·9-s + 0.603·11-s − 0.554·13-s + 1.45·17-s + 1.83·19-s + 0.436·21-s + 0.834·23-s + 0.192·27-s − 1.48·29-s + 0.348·33-s + 1.64·37-s − 0.320·39-s + 0.312·41-s − 1.82·43-s − 3/7·49-s + 0.840·51-s − 1.37·53-s + 1.05·57-s − 0.781·59-s − 0.256·61-s + 0.251·63-s − 0.977·67-s + 0.481·69-s + 0.474·71-s + 0.468·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.126335600\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.126335600\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−7.982162846725378810260215705771, −7.76080264562030159981845189944, −7.08585131058000530141526556595, −6.05970020366042932284231280771, −5.21210384629919748872027405183, −4.67819682927358498497222673211, −3.52252322955467007303473713076, −3.06485371140910972028201106868, −1.81470691842328414938166268450, −1.03991305432921092299789461902,
1.03991305432921092299789461902, 1.81470691842328414938166268450, 3.06485371140910972028201106868, 3.52252322955467007303473713076, 4.67819682927358498497222673211, 5.21210384629919748872027405183, 6.05970020366042932284231280771, 7.08585131058000530141526556595, 7.76080264562030159981845189944, 7.982162846725378810260215705771
