L(s) = 1 | − 1.24·2-s − 0.445·4-s + 2·5-s + 3.04·8-s − 2.49·10-s + 1.24·11-s − 1.10·13-s − 2.91·16-s + 6.09·17-s − 6·19-s − 0.890·20-s − 1.55·22-s − 6.51·23-s − 25-s + 1.38·26-s − 0.198·29-s − 6.27·31-s − 2.46·32-s − 7.60·34-s − 7.46·37-s + 7.48·38-s + 6.09·40-s − 8.59·41-s + 9.25·43-s − 0.554·44-s + 8.12·46-s + 3.28·47-s + ⋯ |
L(s) = 1 | − 0.881·2-s − 0.222·4-s + 0.894·5-s + 1.07·8-s − 0.788·10-s + 0.375·11-s − 0.307·13-s − 0.727·16-s + 1.47·17-s − 1.37·19-s − 0.199·20-s − 0.331·22-s − 1.35·23-s − 0.200·25-s + 0.271·26-s − 0.0367·29-s − 1.12·31-s − 0.436·32-s − 1.30·34-s − 1.22·37-s + 1.21·38-s + 0.964·40-s − 1.34·41-s + 1.41·43-s − 0.0836·44-s + 1.19·46-s + 0.479·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3087 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3087 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + 1.24T + 2T^{2} \) |
| 5 | \( 1 - 2T + 5T^{2} \) |
| 11 | \( 1 - 1.24T + 11T^{2} \) |
| 13 | \( 1 + 1.10T + 13T^{2} \) |
| 17 | \( 1 - 6.09T + 17T^{2} \) |
| 19 | \( 1 + 6T + 19T^{2} \) |
| 23 | \( 1 + 6.51T + 23T^{2} \) |
| 29 | \( 1 + 0.198T + 29T^{2} \) |
| 31 | \( 1 + 6.27T + 31T^{2} \) |
| 37 | \( 1 + 7.46T + 37T^{2} \) |
| 41 | \( 1 + 8.59T + 41T^{2} \) |
| 43 | \( 1 - 9.25T + 43T^{2} \) |
| 47 | \( 1 - 3.28T + 47T^{2} \) |
| 53 | \( 1 - 2.26T + 53T^{2} \) |
| 59 | \( 1 - 1.01T + 59T^{2} \) |
| 61 | \( 1 + 10.7T + 61T^{2} \) |
| 67 | \( 1 + 6.69T + 67T^{2} \) |
| 71 | \( 1 + 0.127T + 71T^{2} \) |
| 73 | \( 1 + 7.82T + 73T^{2} \) |
| 79 | \( 1 - 6.93T + 79T^{2} \) |
| 83 | \( 1 - 11.3T + 83T^{2} \) |
| 89 | \( 1 + 14.8T + 89T^{2} \) |
| 97 | \( 1 - 7.03T + 97T^{2} \) |
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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
−8.426275467183016377997182253191, −7.75987917835222769288494757095, −7.00395174497938949798657969984, −6.00281541613503069565060553481, −5.43485633940292560278441215361, −4.39913719096381879994293015194, −3.57833540478086527453349331561, −2.15029840838319665926459337895, −1.46629063618514410801234760090, 0,
1.46629063618514410801234760090, 2.15029840838319665926459337895, 3.57833540478086527453349331561, 4.39913719096381879994293015194, 5.43485633940292560278441215361, 6.00281541613503069565060553481, 7.00395174497938949798657969984, 7.75987917835222769288494757095, 8.426275467183016377997182253191