L(s) = 1 | + 2-s − 3-s + 4-s − 2.67·5-s − 6-s + 2.79·7-s + 8-s + 9-s − 2.67·10-s − 4.92·11-s − 12-s − 6.47·13-s + 2.79·14-s + 2.67·15-s + 16-s + 2·17-s + 18-s − 19-s − 2.67·20-s − 2.79·21-s − 4.92·22-s − 4.24·23-s − 24-s + 2.15·25-s − 6.47·26-s − 27-s + 2.79·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.19·5-s − 0.408·6-s + 1.05·7-s + 0.353·8-s + 0.333·9-s − 0.846·10-s − 1.48·11-s − 0.288·12-s − 1.79·13-s + 0.746·14-s + 0.690·15-s + 0.250·16-s + 0.485·17-s + 0.235·18-s − 0.229·19-s − 0.598·20-s − 0.609·21-s − 1.04·22-s − 0.885·23-s − 0.204·24-s + 0.431·25-s − 1.26·26-s − 0.192·27-s + 0.527·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.420116201\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.420116201\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| 53 | \( 1 + T \) |
good | 5 | \( 1 + 2.67T + 5T^{2} \) |
| 7 | \( 1 - 2.79T + 7T^{2} \) |
| 11 | \( 1 + 4.92T + 11T^{2} \) |
| 13 | \( 1 + 6.47T + 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 23 | \( 1 + 4.24T + 23T^{2} \) |
| 29 | \( 1 - 6.24T + 29T^{2} \) |
| 31 | \( 1 + 2.73T + 31T^{2} \) |
| 37 | \( 1 - 1.11T + 37T^{2} \) |
| 41 | \( 1 - 10.7T + 41T^{2} \) |
| 43 | \( 1 + 5.71T + 43T^{2} \) |
| 47 | \( 1 - 5.20T + 47T^{2} \) |
| 59 | \( 1 - 0.150T + 59T^{2} \) |
| 61 | \( 1 - 9.72T + 61T^{2} \) |
| 67 | \( 1 - 6.79T + 67T^{2} \) |
| 71 | \( 1 - 12.8T + 71T^{2} \) |
| 73 | \( 1 + 4.26T + 73T^{2} \) |
| 79 | \( 1 + 16.0T + 79T^{2} \) |
| 83 | \( 1 - 15.3T + 83T^{2} \) |
| 89 | \( 1 - 11.5T + 89T^{2} \) |
| 97 | \( 1 - 14.3T + 97T^{2} \) |
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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
−7.79271110974461028278797309984, −7.54940051554267481618418847387, −6.71835464507516490055942680120, −5.62251831017561041738795178112, −5.08305343327476778237127662947, −4.59128999990262245631563240691, −3.91048837253194134815910176999, −2.77895714458371651665794576954, −2.08486854872079991857491740535, −0.55226727858622188062498821173,
0.55226727858622188062498821173, 2.08486854872079991857491740535, 2.77895714458371651665794576954, 3.91048837253194134815910176999, 4.59128999990262245631563240691, 5.08305343327476778237127662947, 5.62251831017561041738795178112, 6.71835464507516490055942680120, 7.54940051554267481618418847387, 7.79271110974461028278797309984