L(s) = 1 | − 2-s + 0.590·3-s + 4-s + 3.96·5-s − 0.590·6-s − 4.53·7-s − 8-s − 2.65·9-s − 3.96·10-s − 5.22·11-s + 0.590·12-s − 0.575·13-s + 4.53·14-s + 2.34·15-s + 16-s − 2.92·17-s + 2.65·18-s + 5.48·19-s + 3.96·20-s − 2.67·21-s + 5.22·22-s + 4.54·23-s − 0.590·24-s + 10.7·25-s + 0.575·26-s − 3.33·27-s − 4.53·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.340·3-s + 0.5·4-s + 1.77·5-s − 0.241·6-s − 1.71·7-s − 0.353·8-s − 0.883·9-s − 1.25·10-s − 1.57·11-s + 0.170·12-s − 0.159·13-s + 1.21·14-s + 0.604·15-s + 0.250·16-s − 0.709·17-s + 0.624·18-s + 1.25·19-s + 0.886·20-s − 0.584·21-s + 1.11·22-s + 0.948·23-s − 0.120·24-s + 2.14·25-s + 0.112·26-s − 0.642·27-s − 0.856·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.299424625\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.299424625\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 2011 | \( 1 - T \) |
good | 3 | \( 1 - 0.590T + 3T^{2} \) |
| 5 | \( 1 - 3.96T + 5T^{2} \) |
| 7 | \( 1 + 4.53T + 7T^{2} \) |
| 11 | \( 1 + 5.22T + 11T^{2} \) |
| 13 | \( 1 + 0.575T + 13T^{2} \) |
| 17 | \( 1 + 2.92T + 17T^{2} \) |
| 19 | \( 1 - 5.48T + 19T^{2} \) |
| 23 | \( 1 - 4.54T + 23T^{2} \) |
| 29 | \( 1 - 1.56T + 29T^{2} \) |
| 31 | \( 1 + 0.117T + 31T^{2} \) |
| 37 | \( 1 - 11.0T + 37T^{2} \) |
| 41 | \( 1 + 9.06T + 41T^{2} \) |
| 43 | \( 1 - 3.86T + 43T^{2} \) |
| 47 | \( 1 + 4.03T + 47T^{2} \) |
| 53 | \( 1 - 4.25T + 53T^{2} \) |
| 59 | \( 1 - 3.55T + 59T^{2} \) |
| 61 | \( 1 - 1.06T + 61T^{2} \) |
| 67 | \( 1 - 2.84T + 67T^{2} \) |
| 71 | \( 1 - 8.41T + 71T^{2} \) |
| 73 | \( 1 - 1.78T + 73T^{2} \) |
| 79 | \( 1 + 6.20T + 79T^{2} \) |
| 83 | \( 1 - 4.25T + 83T^{2} \) |
| 89 | \( 1 - 15.6T + 89T^{2} \) |
| 97 | \( 1 - 6.76T + 97T^{2} \) |
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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
−8.701303505999355665683445559520, −7.78515846435636082936333377559, −6.89315768975240701623837506663, −6.29868437504582895565240508768, −5.62731720120797373432259840065, −5.07094567967195794364333502597, −3.26387440453098128760580690234, −2.74560556430492763947245099556, −2.22630032738932199447365912289, −0.67210535882495559166870142994,
0.67210535882495559166870142994, 2.22630032738932199447365912289, 2.74560556430492763947245099556, 3.26387440453098128760580690234, 5.07094567967195794364333502597, 5.62731720120797373432259840065, 6.29868437504582895565240508768, 6.89315768975240701623837506663, 7.78515846435636082936333377559, 8.701303505999355665683445559520