| L(s) = 1 | − 2.44·2-s − 0.864·3-s + 3.99·4-s + 2.08·5-s + 2.11·6-s − 1.35·7-s − 4.89·8-s − 2.25·9-s − 5.10·10-s + 1.47·11-s − 3.45·12-s + 5.96·13-s + 3.31·14-s − 1.80·15-s + 3.98·16-s − 1.61·17-s + 5.51·18-s − 6.49·19-s + 8.33·20-s + 1.17·21-s − 3.60·22-s + 5.72·23-s + 4.23·24-s − 0.651·25-s − 14.5·26-s + 4.54·27-s − 5.41·28-s + ⋯ |
| L(s) = 1 | − 1.73·2-s − 0.499·3-s + 1.99·4-s + 0.932·5-s + 0.864·6-s − 0.512·7-s − 1.73·8-s − 0.750·9-s − 1.61·10-s + 0.443·11-s − 0.997·12-s + 1.65·13-s + 0.886·14-s − 0.465·15-s + 0.996·16-s − 0.390·17-s + 1.30·18-s − 1.48·19-s + 1.86·20-s + 0.255·21-s − 0.768·22-s + 1.19·23-s + 0.863·24-s − 0.130·25-s − 2.86·26-s + 0.873·27-s − 1.02·28-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.44T + 2T^{2} \) |
| 3 | \( 1 + 0.864T + 3T^{2} \) |
| 5 | \( 1 - 2.08T + 5T^{2} \) |
| 7 | \( 1 + 1.35T + 7T^{2} \) |
| 11 | \( 1 - 1.47T + 11T^{2} \) |
| 13 | \( 1 - 5.96T + 13T^{2} \) |
| 17 | \( 1 + 1.61T + 17T^{2} \) |
| 19 | \( 1 + 6.49T + 19T^{2} \) |
| 23 | \( 1 - 5.72T + 23T^{2} \) |
| 29 | \( 1 - 7.46T + 29T^{2} \) |
| 31 | \( 1 + 9.92T + 31T^{2} \) |
| 37 | \( 1 + 6.38T + 37T^{2} \) |
| 41 | \( 1 - 10.6T + 41T^{2} \) |
| 43 | \( 1 - 6.09T + 43T^{2} \) |
| 47 | \( 1 + 5.75T + 47T^{2} \) |
| 53 | \( 1 + 11.1T + 53T^{2} \) |
| 59 | \( 1 + 0.156T + 59T^{2} \) |
| 61 | \( 1 + 5.80T + 61T^{2} \) |
| 67 | \( 1 - 10.2T + 67T^{2} \) |
| 73 | \( 1 + 9.98T + 73T^{2} \) |
| 79 | \( 1 - 15.2T + 79T^{2} \) |
| 83 | \( 1 + 4.92T + 83T^{2} \) |
| 89 | \( 1 + 7.42T + 89T^{2} \) |
| 97 | \( 1 + 1.44T + 97T^{2} \) |
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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
−7.62250570788441250545982971487, −6.65177907175319376323201861411, −6.29880102430791803858465630405, −5.94530960579299898166478542470, −4.85474355829196138837099615477, −3.65271211654434458089014580363, −2.72607595789144632924411705165, −1.86329580277180244268728108594, −1.07815442080796055076883283545, 0,
1.07815442080796055076883283545, 1.86329580277180244268728108594, 2.72607595789144632924411705165, 3.65271211654434458089014580363, 4.85474355829196138837099615477, 5.94530960579299898166478542470, 6.29880102430791803858465630405, 6.65177907175319376323201861411, 7.62250570788441250545982971487
