| L(s) = 1 | − 2-s − 2.02·3-s + 4-s + 3.49·5-s + 2.02·6-s − 2.15·7-s − 8-s + 1.08·9-s − 3.49·10-s + 0.642·11-s − 2.02·12-s − 2.94·13-s + 2.15·14-s − 7.06·15-s + 16-s + 1.73·17-s − 1.08·18-s + 2.22·19-s + 3.49·20-s + 4.35·21-s − 0.642·22-s + 7.43·23-s + 2.02·24-s + 7.23·25-s + 2.94·26-s + 3.86·27-s − 2.15·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.16·3-s + 0.5·4-s + 1.56·5-s + 0.825·6-s − 0.814·7-s − 0.353·8-s + 0.362·9-s − 1.10·10-s + 0.193·11-s − 0.583·12-s − 0.816·13-s + 0.575·14-s − 1.82·15-s + 0.250·16-s + 0.419·17-s − 0.256·18-s + 0.510·19-s + 0.782·20-s + 0.950·21-s − 0.137·22-s + 1.54·23-s + 0.412·24-s + 1.44·25-s + 0.577·26-s + 0.744·27-s − 0.407·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.040068532\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.040068532\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 4021 | \( 1+O(T) \) |
| good | 3 | \( 1 + 2.02T + 3T^{2} \) |
| 5 | \( 1 - 3.49T + 5T^{2} \) |
| 7 | \( 1 + 2.15T + 7T^{2} \) |
| 11 | \( 1 - 0.642T + 11T^{2} \) |
| 13 | \( 1 + 2.94T + 13T^{2} \) |
| 17 | \( 1 - 1.73T + 17T^{2} \) |
| 19 | \( 1 - 2.22T + 19T^{2} \) |
| 23 | \( 1 - 7.43T + 23T^{2} \) |
| 29 | \( 1 + 4.41T + 29T^{2} \) |
| 31 | \( 1 + 1.19T + 31T^{2} \) |
| 37 | \( 1 - 0.585T + 37T^{2} \) |
| 41 | \( 1 - 9.41T + 41T^{2} \) |
| 43 | \( 1 - 0.549T + 43T^{2} \) |
| 47 | \( 1 - 4.47T + 47T^{2} \) |
| 53 | \( 1 + 0.998T + 53T^{2} \) |
| 59 | \( 1 - 1.72T + 59T^{2} \) |
| 61 | \( 1 + 2.52T + 61T^{2} \) |
| 67 | \( 1 + 7.72T + 67T^{2} \) |
| 71 | \( 1 - 1.70T + 71T^{2} \) |
| 73 | \( 1 + 10.7T + 73T^{2} \) |
| 79 | \( 1 + 4.59T + 79T^{2} \) |
| 83 | \( 1 - 4.17T + 83T^{2} \) |
| 89 | \( 1 + 12.7T + 89T^{2} \) |
| 97 | \( 1 + 8.89T + 97T^{2} \) |
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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
−7.56840722032731315939554383575, −7.04771202265751330918681566281, −6.36695350752280345396107979309, −5.80164824138072437964107424497, −5.41312408058609827507144110909, −4.59624479260497638381535978914, −3.18549952622391428937479588317, −2.55251199195675400186557282256, −1.50438732344014887828913473927, −0.62129114487014635899896715161,
0.62129114487014635899896715161, 1.50438732344014887828913473927, 2.55251199195675400186557282256, 3.18549952622391428937479588317, 4.59624479260497638381535978914, 5.41312408058609827507144110909, 5.80164824138072437964107424497, 6.36695350752280345396107979309, 7.04771202265751330918681566281, 7.56840722032731315939554383575
