| L(s) = 1 | − 1.99·2-s − 1.58·3-s + 1.97·4-s − 2.66·5-s + 3.16·6-s − 1.73·7-s + 0.0439·8-s − 0.476·9-s + 5.32·10-s + 2.93·11-s − 3.14·12-s + 3.46·14-s + 4.23·15-s − 4.04·16-s + 1.90·17-s + 0.951·18-s + 4.24·19-s − 5.27·20-s + 2.76·21-s − 5.86·22-s − 3.75·23-s − 0.0697·24-s + 2.11·25-s + 5.52·27-s − 3.44·28-s − 4.27·29-s − 8.45·30-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 0.917·3-s + 0.988·4-s − 1.19·5-s + 1.29·6-s − 0.657·7-s + 0.0155·8-s − 0.158·9-s + 1.68·10-s + 0.886·11-s − 0.906·12-s + 0.927·14-s + 1.09·15-s − 1.01·16-s + 0.461·17-s + 0.224·18-s + 0.974·19-s − 1.17·20-s + 0.602·21-s − 1.24·22-s − 0.783·23-s − 0.0142·24-s + 0.423·25-s + 1.06·27-s − 0.650·28-s − 0.793·29-s − 1.54·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5239 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5239 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2846726949\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2846726949\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 \) |
| 31 | \( 1 + T \) |
| good | 2 | \( 1 + 1.99T + 2T^{2} \) |
| 3 | \( 1 + 1.58T + 3T^{2} \) |
| 5 | \( 1 + 2.66T + 5T^{2} \) |
| 7 | \( 1 + 1.73T + 7T^{2} \) |
| 11 | \( 1 - 2.93T + 11T^{2} \) |
| 17 | \( 1 - 1.90T + 17T^{2} \) |
| 19 | \( 1 - 4.24T + 19T^{2} \) |
| 23 | \( 1 + 3.75T + 23T^{2} \) |
| 29 | \( 1 + 4.27T + 29T^{2} \) |
| 37 | \( 1 - 3.05T + 37T^{2} \) |
| 41 | \( 1 - 5.80T + 41T^{2} \) |
| 43 | \( 1 - 11.6T + 43T^{2} \) |
| 47 | \( 1 - 10.5T + 47T^{2} \) |
| 53 | \( 1 + 13.8T + 53T^{2} \) |
| 59 | \( 1 - 13.7T + 59T^{2} \) |
| 61 | \( 1 - 5.05T + 61T^{2} \) |
| 67 | \( 1 + 12.7T + 67T^{2} \) |
| 71 | \( 1 + 5.83T + 71T^{2} \) |
| 73 | \( 1 + 4.92T + 73T^{2} \) |
| 79 | \( 1 - 0.743T + 79T^{2} \) |
| 83 | \( 1 - 0.165T + 83T^{2} \) |
| 89 | \( 1 - 1.85T + 89T^{2} \) |
| 97 | \( 1 + 6.35T + 97T^{2} \) |
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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
−8.129562766829407621586498073020, −7.58070928390459707107172268892, −7.04718478512620873799016497146, −6.18931902162715985604415791820, −5.56447780353112846403268900253, −4.39288421892178720316187932877, −3.77846271767058957360363562499, −2.70768814679376052831796159372, −1.26221692554604199773089382613, −0.42833054753345948796124086597,
0.42833054753345948796124086597, 1.26221692554604199773089382613, 2.70768814679376052831796159372, 3.77846271767058957360363562499, 4.39288421892178720316187932877, 5.56447780353112846403268900253, 6.18931902162715985604415791820, 7.04718478512620873799016497146, 7.58070928390459707107172268892, 8.129562766829407621586498073020
