| L(s) = 1 | + 2-s + 3.30·3-s + 4-s − 3.07·5-s + 3.30·6-s − 4.11·7-s + 8-s + 7.89·9-s − 3.07·10-s − 4.16·11-s + 3.30·12-s − 6.00·13-s − 4.11·14-s − 10.1·15-s + 16-s + 4.19·17-s + 7.89·18-s − 19-s − 3.07·20-s − 13.5·21-s − 4.16·22-s + 7.28·23-s + 3.30·24-s + 4.47·25-s − 6.00·26-s + 16.1·27-s − 4.11·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.90·3-s + 0.5·4-s − 1.37·5-s + 1.34·6-s − 1.55·7-s + 0.353·8-s + 2.63·9-s − 0.973·10-s − 1.25·11-s + 0.952·12-s − 1.66·13-s − 1.09·14-s − 2.62·15-s + 0.250·16-s + 1.01·17-s + 1.86·18-s − 0.229·19-s − 0.688·20-s − 2.96·21-s − 0.888·22-s + 1.51·23-s + 0.673·24-s + 0.895·25-s − 1.17·26-s + 3.10·27-s − 0.776·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.905602116\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.905602116\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| 211 | \( 1 + T \) |
| good | 3 | \( 1 - 3.30T + 3T^{2} \) |
| 5 | \( 1 + 3.07T + 5T^{2} \) |
| 7 | \( 1 + 4.11T + 7T^{2} \) |
| 11 | \( 1 + 4.16T + 11T^{2} \) |
| 13 | \( 1 + 6.00T + 13T^{2} \) |
| 17 | \( 1 - 4.19T + 17T^{2} \) |
| 23 | \( 1 - 7.28T + 23T^{2} \) |
| 29 | \( 1 - 9.67T + 29T^{2} \) |
| 31 | \( 1 + 2.11T + 31T^{2} \) |
| 37 | \( 1 - 8.60T + 37T^{2} \) |
| 41 | \( 1 - 11.1T + 41T^{2} \) |
| 43 | \( 1 - 6.48T + 43T^{2} \) |
| 47 | \( 1 - 8.80T + 47T^{2} \) |
| 53 | \( 1 + 11.6T + 53T^{2} \) |
| 59 | \( 1 - 11.2T + 59T^{2} \) |
| 61 | \( 1 + 5.32T + 61T^{2} \) |
| 67 | \( 1 - 2.92T + 67T^{2} \) |
| 71 | \( 1 + 1.20T + 71T^{2} \) |
| 73 | \( 1 + 9.70T + 73T^{2} \) |
| 79 | \( 1 + 8.78T + 79T^{2} \) |
| 83 | \( 1 - 2.66T + 83T^{2} \) |
| 89 | \( 1 + 5.01T + 89T^{2} \) |
| 97 | \( 1 + 7.76T + 97T^{2} \) |
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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
−7.77979110763607978201112838746, −7.25406787440519511552139517174, −6.88248358283947036006233228267, −5.65283821388291352427094534186, −4.56266153419286481808722147163, −4.21998180179090896836618566873, −3.24854305146077828315408511085, −2.77417440498180207615663927005, −2.61457500574900883160080148402, −0.76665021211034476944058435062,
0.76665021211034476944058435062, 2.61457500574900883160080148402, 2.77417440498180207615663927005, 3.24854305146077828315408511085, 4.21998180179090896836618566873, 4.56266153419286481808722147163, 5.65283821388291352427094534186, 6.88248358283947036006233228267, 7.25406787440519511552139517174, 7.77979110763607978201112838746
