| L(s) = 1 | + 3-s − 7-s + 9-s + 11-s − 17-s + 19-s − 21-s + 23-s + 27-s + 29-s + 31-s + 33-s + 37-s − 41-s + 43-s − 47-s + 49-s − 51-s − 53-s + 57-s + 59-s + 61-s − 63-s − 67-s + 69-s + 71-s + 73-s + ⋯ |
| L(s) = 1 | + 3-s − 7-s + 9-s + 11-s − 17-s + 19-s − 21-s + 23-s + 27-s + 29-s + 31-s + 33-s + 37-s − 41-s + 43-s − 47-s + 49-s − 51-s − 53-s + 57-s + 59-s + 61-s − 63-s − 67-s + 69-s + 71-s + 73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.794686369\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.794686369\) |
| \(L(1)\) |
\(\approx\) |
\(1.558666443\) |
| \(L(1)\) |
\(\approx\) |
\(1.558666443\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 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
−25.5027612389355468151327572607, −24.994760721560700353121042288497, −24.071591513501749663325852024363, −22.74387573155367473213811132884, −22.050714858996110861389444396315, −20.9796305837592956606137692630, −19.89332147839737164374480733280, −19.49669050114720999334750737454, −18.50283051060942508142848627382, −17.318546324845978916304751932761, −16.1270309622793331006883390254, −15.41339638944074275652689958453, −14.33694089080103506974825609766, −13.49190401906746977627230903837, −12.66436734651884907684905757019, −11.47762159524207867349435111946, −10.03554642324674520303253140194, −9.317625338618921657497391409504, −8.460735156289091030179115038193, −7.11702920681214076217149197981, −6.38198552586259302660607619646, −4.63782467362693299284394452443, −3.518156196496437855573716749134, −2.60072801849673366923250252001, −1.05325893699922446326153868033,
1.05325893699922446326153868033, 2.60072801849673366923250252001, 3.518156196496437855573716749134, 4.63782467362693299284394452443, 6.38198552586259302660607619646, 7.11702920681214076217149197981, 8.460735156289091030179115038193, 9.317625338618921657497391409504, 10.03554642324674520303253140194, 11.47762159524207867349435111946, 12.66436734651884907684905757019, 13.49190401906746977627230903837, 14.33694089080103506974825609766, 15.41339638944074275652689958453, 16.1270309622793331006883390254, 17.318546324845978916304751932761, 18.50283051060942508142848627382, 19.49669050114720999334750737454, 19.89332147839737164374480733280, 20.9796305837592956606137692630, 22.050714858996110861389444396315, 22.74387573155367473213811132884, 24.071591513501749663325852024363, 24.994760721560700353121042288497, 25.5027612389355468151327572607
