L(s) = 1 | − 2.79·2-s − 2.79·3-s + 5.79·4-s + 3.79·5-s + 7.79·6-s + 7-s − 10.5·8-s + 4.79·9-s − 10.5·10-s − 0.791·11-s − 16.1·12-s + 13-s − 2.79·14-s − 10.5·15-s + 17.9·16-s − 0.791·17-s − 13.3·18-s + 0.582·19-s + 21.9·20-s − 2.79·21-s + 2.20·22-s − 23-s + 29.5·24-s + 9.37·25-s − 2.79·26-s − 4.99·27-s + 5.79·28-s + ⋯ |
L(s) = 1 | − 1.97·2-s − 1.61·3-s + 2.89·4-s + 1.69·5-s + 3.18·6-s + 0.377·7-s − 3.74·8-s + 1.59·9-s − 3.34·10-s − 0.238·11-s − 4.66·12-s + 0.277·13-s − 0.746·14-s − 2.73·15-s + 4.48·16-s − 0.191·17-s − 3.15·18-s + 0.133·19-s + 4.90·20-s − 0.609·21-s + 0.470·22-s − 0.208·23-s + 6.02·24-s + 1.87·25-s − 0.547·26-s − 0.962·27-s + 1.09·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 299 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 299 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4569990983\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4569990983\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 13 | \( 1 - T \) |
| 23 | \( 1 + T \) |
good | 2 | \( 1 + 2.79T + 2T^{2} \) |
| 3 | \( 1 + 2.79T + 3T^{2} \) |
| 5 | \( 1 - 3.79T + 5T^{2} \) |
| 7 | \( 1 - T + 7T^{2} \) |
| 11 | \( 1 + 0.791T + 11T^{2} \) |
| 17 | \( 1 + 0.791T + 17T^{2} \) |
| 19 | \( 1 - 0.582T + 19T^{2} \) |
| 29 | \( 1 + 3.20T + 29T^{2} \) |
| 31 | \( 1 - 3.20T + 31T^{2} \) |
| 37 | \( 1 - 11.5T + 37T^{2} \) |
| 41 | \( 1 - 0.582T + 41T^{2} \) |
| 43 | \( 1 + 2.37T + 43T^{2} \) |
| 47 | \( 1 - 3.79T + 47T^{2} \) |
| 53 | \( 1 - 7T + 53T^{2} \) |
| 59 | \( 1 + 4.16T + 59T^{2} \) |
| 61 | \( 1 - 5T + 61T^{2} \) |
| 67 | \( 1 - 5T + 67T^{2} \) |
| 71 | \( 1 - 10.5T + 71T^{2} \) |
| 73 | \( 1 - 8.20T + 73T^{2} \) |
| 79 | \( 1 + 3.79T + 79T^{2} \) |
| 83 | \( 1 - 12.9T + 83T^{2} \) |
| 89 | \( 1 - 11.5T + 89T^{2} \) |
| 97 | \( 1 + 4.58T + 97T^{2} \) |
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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
−11.22113375094048685682718852660, −10.68224783128585793691876666740, −9.902096644067853016212323238474, −9.289028035625784724535726691343, −8.010030022447235861942561285473, −6.74816789051442958766134962695, −6.13606497772836372235964294502, −5.35043650678296737410816224885, −2.26581472449932792214633875264, −1.04716503990335394306377205145,
1.04716503990335394306377205145, 2.26581472449932792214633875264, 5.35043650678296737410816224885, 6.13606497772836372235964294502, 6.74816789051442958766134962695, 8.010030022447235861942561285473, 9.289028035625784724535726691343, 9.902096644067853016212323238474, 10.68224783128585793691876666740, 11.22113375094048685682718852660