| L(s) = 1 | + 3-s + 5-s − 7-s + 9-s − 11-s − 13-s + 15-s + 17-s − 19-s − 21-s − 23-s + 25-s + 27-s + 29-s + 31-s − 33-s − 35-s − 37-s − 39-s + 41-s + 43-s + 45-s + 47-s + 49-s + 51-s + 53-s − 55-s + ⋯ |
| L(s) = 1 | + 3-s + 5-s − 7-s + 9-s − 11-s − 13-s + 15-s + 17-s − 19-s − 21-s − 23-s + 25-s + 27-s + 29-s + 31-s − 33-s − 35-s − 37-s − 39-s + 41-s + 43-s + 45-s + 47-s + 49-s + 51-s + 53-s − 55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1016 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1016 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(3.253924112\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.253924112\) |
| \(L(1)\) |
\(\approx\) |
\(1.576968438\) |
| \(L(1)\) |
\(\approx\) |
\(1.576968438\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 127 | \( 1 \) |
| good | 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
−21.25268286503159065291065915081, −20.83015178093231663366035065935, −19.73027985886277748336545084921, −19.206069508757536642599765796007, −18.449442312018656577891270795770, −17.56674527490329537448861824173, −16.66397675715987317123789057248, −15.82821361137024094531375485637, −15.09380646939201568405400529756, −14.05376740125206652368517299513, −13.73627178102414991365637806567, −12.55965662131605612717761310999, −12.45660558807179424992551889288, −10.46030921063116868207070500039, −10.11489129865145873889970328277, −9.4446156714605272902815197354, −8.51283892299492366080687750596, −7.6371565968330169914344907618, −6.73119235162422725411456044377, −5.841389240415433389151076876739, −4.8302493695832161705020716109, −3.70409332165738140536478928533, −2.567353241348533321313266159994, −2.319428013873288434997152834098, −0.77476751040949914418414486720,
0.77476751040949914418414486720, 2.319428013873288434997152834098, 2.567353241348533321313266159994, 3.70409332165738140536478928533, 4.8302493695832161705020716109, 5.841389240415433389151076876739, 6.73119235162422725411456044377, 7.6371565968330169914344907618, 8.51283892299492366080687750596, 9.4446156714605272902815197354, 10.11489129865145873889970328277, 10.46030921063116868207070500039, 12.45660558807179424992551889288, 12.55965662131605612717761310999, 13.73627178102414991365637806567, 14.05376740125206652368517299513, 15.09380646939201568405400529756, 15.82821361137024094531375485637, 16.66397675715987317123789057248, 17.56674527490329537448861824173, 18.449442312018656577891270795770, 19.206069508757536642599765796007, 19.73027985886277748336545084921, 20.83015178093231663366035065935, 21.25268286503159065291065915081
