| L(s) = 1 | + 3-s + 7-s + 9-s − 2·13-s − 2·17-s + 21-s + 27-s + 6·29-s + 4·31-s + 2·37-s − 2·39-s + 10·41-s − 4·43-s + 49-s − 2·51-s − 2·53-s + 4·59-s + 6·61-s + 63-s + 12·67-s + 12·71-s + 2·73-s + 81-s − 4·83-s + 6·87-s + 10·89-s − 2·91-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.377·7-s + 1/3·9-s − 0.554·13-s − 0.485·17-s + 0.218·21-s + 0.192·27-s + 1.11·29-s + 0.718·31-s + 0.328·37-s − 0.320·39-s + 1.56·41-s − 0.609·43-s + 1/7·49-s − 0.280·51-s − 0.274·53-s + 0.520·59-s + 0.768·61-s + 0.125·63-s + 1.46·67-s + 1.42·71-s + 0.234·73-s + 1/9·81-s − 0.439·83-s + 0.643·87-s + 1.05·89-s − 0.209·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.482712218\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.482712218\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 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
−8.283696226298228974200537349004, −7.86798921225006925810451158422, −6.96145898552098178367844801540, −6.36754999445596353007197639706, −5.30172594581232812650816771339, −4.60206911844341426469725578390, −3.85627367108123214807701690723, −2.79179785063727029498459105260, −2.13802530746157917992691917276, −0.881609998717120883445066944292,
0.881609998717120883445066944292, 2.13802530746157917992691917276, 2.79179785063727029498459105260, 3.85627367108123214807701690723, 4.60206911844341426469725578390, 5.30172594581232812650816771339, 6.36754999445596353007197639706, 6.96145898552098178367844801540, 7.86798921225006925810451158422, 8.283696226298228974200537349004
