| L(s) = 1 | + 0.732·3-s + 5-s − 2.46·9-s − 2·11-s − 2.73·13-s + 0.732·15-s + 0.535·17-s − 19-s + 5.46·23-s + 25-s − 4·27-s − 3.46·29-s + 4·31-s − 1.46·33-s + 9.66·37-s − 2·39-s + 7.46·41-s − 10.9·43-s − 2.46·45-s + 10.9·47-s − 7·49-s + 0.392·51-s − 5.66·53-s − 2·55-s − 0.732·57-s + 5.46·59-s + 13.4·61-s + ⋯ |
| L(s) = 1 | + 0.422·3-s + 0.447·5-s − 0.821·9-s − 0.603·11-s − 0.757·13-s + 0.189·15-s + 0.129·17-s − 0.229·19-s + 1.13·23-s + 0.200·25-s − 0.769·27-s − 0.643·29-s + 0.718·31-s − 0.254·33-s + 1.58·37-s − 0.320·39-s + 1.16·41-s − 1.66·43-s − 0.367·45-s + 1.59·47-s − 49-s + 0.0549·51-s − 0.777·53-s − 0.269·55-s − 0.0969·57-s + 0.711·59-s + 1.72·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.022312240\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.022312240\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - 0.732T + 3T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 + 2.73T + 13T^{2} \) |
| 17 | \( 1 - 0.535T + 17T^{2} \) |
| 23 | \( 1 - 5.46T + 23T^{2} \) |
| 29 | \( 1 + 3.46T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 - 9.66T + 37T^{2} \) |
| 41 | \( 1 - 7.46T + 41T^{2} \) |
| 43 | \( 1 + 10.9T + 43T^{2} \) |
| 47 | \( 1 - 10.9T + 47T^{2} \) |
| 53 | \( 1 + 5.66T + 53T^{2} \) |
| 59 | \( 1 - 5.46T + 59T^{2} \) |
| 61 | \( 1 - 13.4T + 61T^{2} \) |
| 67 | \( 1 + 6.19T + 67T^{2} \) |
| 71 | \( 1 + 2.92T + 71T^{2} \) |
| 73 | \( 1 - 10.3T + 73T^{2} \) |
| 79 | \( 1 - 12.3T + 79T^{2} \) |
| 83 | \( 1 + 1.46T + 83T^{2} \) |
| 89 | \( 1 - 3.46T + 89T^{2} \) |
| 97 | \( 1 - 1.66T + 97T^{2} \) |
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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
−8.069181190337097198490058876220, −7.49245561049059689975223088971, −6.65565177181366095152095799586, −5.88449958217790189506522001377, −5.22380192212532715203088631149, −4.54883780686391483599950672962, −3.43729941510657946948796731233, −2.69416474803842986388366058074, −2.11933650144255510014304664064, −0.70562099199354715333591481619,
0.70562099199354715333591481619, 2.11933650144255510014304664064, 2.69416474803842986388366058074, 3.43729941510657946948796731233, 4.54883780686391483599950672962, 5.22380192212532715203088631149, 5.88449958217790189506522001377, 6.65565177181366095152095799586, 7.49245561049059689975223088971, 8.069181190337097198490058876220
