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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1436 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1436 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.481810395\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.481810395\) |
\(L(1)\) |
\(\approx\) |
\(1.041718488\) |
\(L(1)\) |
\(\approx\) |
\(1.041718488\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 359 | \( 1 \) |
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
−21.05491217444651484613203822905, −20.20035018852979765655085598102, −18.91506320801277222816481669756, −18.19613112458161981207758941615, −17.78415869913353712149326404416, −17.07864318969196592496815640714, −16.40275443761754175685360569040, −15.549662691084442132445616177610, −14.5422445534566132769263245271, −13.959527154942311854536104177516, −12.9701941917987534194556495938, −12.31096837953157275789414997717, −11.487818800298360713865851819011, −10.74006442528658566730387389585, −9.9176246973379649777448196309, −9.51255207883783691564611278522, −7.92469231841933224369673310845, −7.56545633175025460009756144070, −6.38531211264394885217226832634, −5.42978343987060621122283746670, −5.227895284182962241292828941, −4.241686668135536140722625307000, −2.74373726140052496244780264255, −1.86601111356161335063962543279, −0.88632416021074644498599849723,
0.88632416021074644498599849723, 1.86601111356161335063962543279, 2.74373726140052496244780264255, 4.241686668135536140722625307000, 5.227895284182962241292828941, 5.42978343987060621122283746670, 6.38531211264394885217226832634, 7.56545633175025460009756144070, 7.92469231841933224369673310845, 9.51255207883783691564611278522, 9.9176246973379649777448196309, 10.74006442528658566730387389585, 11.487818800298360713865851819011, 12.31096837953157275789414997717, 12.9701941917987534194556495938, 13.959527154942311854536104177516, 14.5422445534566132769263245271, 15.549662691084442132445616177610, 16.40275443761754175685360569040, 17.07864318969196592496815640714, 17.78415869913353712149326404416, 18.19613112458161981207758941615, 18.91506320801277222816481669756, 20.20035018852979765655085598102, 21.05491217444651484613203822905