L(s) = 1 | − 1.73·2-s + 2.37·3-s + 1.01·4-s − 4.11·6-s + 2.01·7-s + 1.70·8-s + 2.62·9-s − 3.39·11-s + 2.41·12-s − 5.63·13-s − 3.49·14-s − 4.99·16-s − 0.866·17-s − 4.55·18-s + 2.46·19-s + 4.76·21-s + 5.89·22-s + 6.37·23-s + 4.05·24-s + 9.79·26-s − 0.892·27-s + 2.04·28-s − 4.52·29-s − 3.51·31-s + 5.26·32-s − 8.05·33-s + 1.50·34-s + ⋯ |
L(s) = 1 | − 1.22·2-s + 1.36·3-s + 0.508·4-s − 1.68·6-s + 0.759·7-s + 0.603·8-s + 0.874·9-s − 1.02·11-s + 0.695·12-s − 1.56·13-s − 0.933·14-s − 1.24·16-s − 0.210·17-s − 1.07·18-s + 0.565·19-s + 1.04·21-s + 1.25·22-s + 1.33·23-s + 0.826·24-s + 1.91·26-s − 0.171·27-s + 0.386·28-s − 0.839·29-s − 0.631·31-s + 0.931·32-s − 1.40·33-s + 0.258·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 241 | \( 1 + T \) |
good | 2 | \( 1 + 1.73T + 2T^{2} \) |
| 3 | \( 1 - 2.37T + 3T^{2} \) |
| 7 | \( 1 - 2.01T + 7T^{2} \) |
| 11 | \( 1 + 3.39T + 11T^{2} \) |
| 13 | \( 1 + 5.63T + 13T^{2} \) |
| 17 | \( 1 + 0.866T + 17T^{2} \) |
| 19 | \( 1 - 2.46T + 19T^{2} \) |
| 23 | \( 1 - 6.37T + 23T^{2} \) |
| 29 | \( 1 + 4.52T + 29T^{2} \) |
| 31 | \( 1 + 3.51T + 31T^{2} \) |
| 37 | \( 1 - 5.19T + 37T^{2} \) |
| 41 | \( 1 - 1.35T + 41T^{2} \) |
| 43 | \( 1 + 8.49T + 43T^{2} \) |
| 47 | \( 1 - 9.44T + 47T^{2} \) |
| 53 | \( 1 + 9.71T + 53T^{2} \) |
| 59 | \( 1 - 6.03T + 59T^{2} \) |
| 61 | \( 1 - 4.45T + 61T^{2} \) |
| 67 | \( 1 - 10.9T + 67T^{2} \) |
| 71 | \( 1 + 3.01T + 71T^{2} \) |
| 73 | \( 1 - 0.255T + 73T^{2} \) |
| 79 | \( 1 + 10.3T + 79T^{2} \) |
| 83 | \( 1 - 16.8T + 83T^{2} \) |
| 89 | \( 1 + 17.4T + 89T^{2} \) |
| 97 | \( 1 + 0.273T + 97T^{2} \) |
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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
−7.85597771402575344655649581680, −7.47094555812268275016669332737, −6.90388147386676771306516765865, −5.31770471834814322012216312978, −4.90609689019943834485586877600, −3.91690973256586203223628602336, −2.77917796061170350019401807369, −2.31185049944708379447400409459, −1.38475331285569718575612177361, 0,
1.38475331285569718575612177361, 2.31185049944708379447400409459, 2.77917796061170350019401807369, 3.91690973256586203223628602336, 4.90609689019943834485586877600, 5.31770471834814322012216312978, 6.90388147386676771306516765865, 7.47094555812268275016669332737, 7.85597771402575344655649581680