L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s − 7-s − 8-s + 9-s + 10-s + 11-s − 12-s + 13-s + 14-s + 15-s + 16-s − 17-s − 18-s + 19-s − 20-s + 21-s − 22-s + 23-s + 24-s + 25-s − 26-s − 27-s − 28-s + ⋯ |
L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s − 7-s − 8-s + 9-s + 10-s + 11-s − 12-s + 13-s + 14-s + 15-s + 16-s − 17-s − 18-s + 19-s − 20-s + 21-s − 22-s + 23-s + 24-s + 25-s − 26-s − 27-s − 28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1013 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1013 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4279146528\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4279146528\) |
\(L(1)\) |
\(\approx\) |
\(0.4290375233\) |
\(L(1)\) |
\(\approx\) |
\(0.4290375233\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 1013 | \( 1 \) |
good | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| 31 | \( 1 - T \) |
| 37 | \( 1 - T \) |
| 41 | \( 1 - T \) |
| 43 | \( 1 + T \) |
| 47 | \( 1 - T \) |
| 53 | \( 1 + T \) |
| 59 | \( 1 - T \) |
| 61 | \( 1 - T \) |
| 67 | \( 1 - T \) |
| 71 | \( 1 + T \) |
| 73 | \( 1 + T \) |
| 79 | \( 1 + T \) |
| 83 | \( 1 + T \) |
| 89 | \( 1 + T \) |
| 97 | \( 1 + T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−21.71013509891824159994121549216, −20.546727842136064607310763249, −19.89652290619082733599250931158, −19.11600110826625612162349408927, −18.536757497188753844101698322290, −17.76534817309231194380028241922, −16.720860022197530990505683766027, −16.409161642340478920723313730, −15.59654275579086545593411016586, −15.081274657105066165937692527434, −13.47417713990504027421814690919, −12.51172679907626270733757152774, −11.83557890960561129679187773443, −11.10819184889596471454704409012, −10.617539060379001899023475022881, −9.33745725015286866263410468933, −8.96641817212610903667197949948, −7.647795427684475806187134151392, −6.86019300420956206350674734107, −6.41167061668114460215750157763, −5.31099077184320002150698348429, −3.90550743947475962169522577397, −3.28454039928955363641087523672, −1.613687908800275422036005715113, −0.59358790904296108818752068785,
0.59358790904296108818752068785, 1.613687908800275422036005715113, 3.28454039928955363641087523672, 3.90550743947475962169522577397, 5.31099077184320002150698348429, 6.41167061668114460215750157763, 6.86019300420956206350674734107, 7.647795427684475806187134151392, 8.96641817212610903667197949948, 9.33745725015286866263410468933, 10.617539060379001899023475022881, 11.10819184889596471454704409012, 11.83557890960561129679187773443, 12.51172679907626270733757152774, 13.47417713990504027421814690919, 15.081274657105066165937692527434, 15.59654275579086545593411016586, 16.409161642340478920723313730, 16.720860022197530990505683766027, 17.76534817309231194380028241922, 18.536757497188753844101698322290, 19.11600110826625612162349408927, 19.89652290619082733599250931158, 20.546727842136064607310763249, 21.71013509891824159994121549216