| L(s) = 1 | − 1.50·2-s − 2.15·3-s + 0.265·4-s − 0.274·5-s + 3.24·6-s − 3.10·7-s + 2.61·8-s + 1.65·9-s + 0.413·10-s − 0.546·11-s − 0.572·12-s + 0.316·13-s + 4.67·14-s + 0.592·15-s − 4.46·16-s + 6.19·17-s − 2.49·18-s + 0.859·19-s − 0.0728·20-s + 6.70·21-s + 0.822·22-s − 3.62·23-s − 5.63·24-s − 4.92·25-s − 0.477·26-s + 2.90·27-s − 0.824·28-s + ⋯ |
| L(s) = 1 | − 1.06·2-s − 1.24·3-s + 0.132·4-s − 0.122·5-s + 1.32·6-s − 1.17·7-s + 0.923·8-s + 0.551·9-s + 0.130·10-s − 0.164·11-s − 0.165·12-s + 0.0879·13-s + 1.24·14-s + 0.153·15-s − 1.11·16-s + 1.50·17-s − 0.587·18-s + 0.197·19-s − 0.0162·20-s + 1.46·21-s + 0.175·22-s − 0.755·23-s − 1.14·24-s − 0.984·25-s − 0.0935·26-s + 0.558·27-s − 0.155·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8023 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3758999297\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3758999297\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 71 | \( 1 - T \) |
| 113 | \( 1 + T \) |
| good | 2 | \( 1 + 1.50T + 2T^{2} \) |
| 3 | \( 1 + 2.15T + 3T^{2} \) |
| 5 | \( 1 + 0.274T + 5T^{2} \) |
| 7 | \( 1 + 3.10T + 7T^{2} \) |
| 11 | \( 1 + 0.546T + 11T^{2} \) |
| 13 | \( 1 - 0.316T + 13T^{2} \) |
| 17 | \( 1 - 6.19T + 17T^{2} \) |
| 19 | \( 1 - 0.859T + 19T^{2} \) |
| 23 | \( 1 + 3.62T + 23T^{2} \) |
| 29 | \( 1 - 5.63T + 29T^{2} \) |
| 31 | \( 1 + 5.22T + 31T^{2} \) |
| 37 | \( 1 - 11.9T + 37T^{2} \) |
| 41 | \( 1 - 8.57T + 41T^{2} \) |
| 43 | \( 1 - 7.71T + 43T^{2} \) |
| 47 | \( 1 + 3.18T + 47T^{2} \) |
| 53 | \( 1 + 10.3T + 53T^{2} \) |
| 59 | \( 1 - 1.09T + 59T^{2} \) |
| 61 | \( 1 - 10.1T + 61T^{2} \) |
| 67 | \( 1 + 1.09T + 67T^{2} \) |
| 73 | \( 1 + 6.15T + 73T^{2} \) |
| 79 | \( 1 - 8.78T + 79T^{2} \) |
| 83 | \( 1 + 12.1T + 83T^{2} \) |
| 89 | \( 1 + 0.385T + 89T^{2} \) |
| 97 | \( 1 - 18.2T + 97T^{2} \) |
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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
−7.69200959126467396120444696405, −7.39850550674577987477776315338, −6.22430370373296534148867827868, −6.06467961136581130549528390259, −5.20383351947842222685968376515, −4.35679052809601592305613024831, −3.55674573088554096141145338511, −2.51894301932025143967823651759, −1.17392476391832507589123958778, −0.45424579951203104182828716283,
0.45424579951203104182828716283, 1.17392476391832507589123958778, 2.51894301932025143967823651759, 3.55674573088554096141145338511, 4.35679052809601592305613024831, 5.20383351947842222685968376515, 6.06467961136581130549528390259, 6.22430370373296534148867827868, 7.39850550674577987477776315338, 7.69200959126467396120444696405
