| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8484402525\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8484402525\) |
| \(L(1)\) |
\(\approx\) |
\(0.9383101982\) |
| \(L(1)\) |
\(\approx\) |
\(0.9383101982\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 61 | \( 1 \) |
| 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 | \( 1 - T \) |
| 67 | \( 1 - T \) |
| 71 | \( 1 - T \) |
| 73 | \( 1 + T \) |
| 79 | \( 1 - T \) |
| 83 | \( 1 + T \) |
| 89 | \( 1 - T \) |
| 97 | \( 1 + T \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−32.82573411853098146644267018675, −31.40500258932120590373403663653, −30.07959899261099683320680980451, −29.098555856325775051405658006562, −28.223749692000577024700664824476, −26.44569956073799372286794402195, −26.05503235862351310609443033328, −25.17795901381565984513361748684, −24.07370651255462611318217007938, −22.03419632198522342627092944428, −20.78392187781689933742302267641, −20.057441138757320313363140006188, −18.672598721958749196534732624781, −18.00576347603514123546512991036, −16.29315012627397787134601995636, −15.489382874162138740862518318787, −13.789131049820828862752138411306, −12.8133754336178107707525794242, −10.669447748894294158852750249409, −9.64611657141885916446348243181, −8.79080826384926706085041087801, −7.32744087186156568040361758365, −5.970300793360114507675288812948, −3.22773639167905402892302632100, −1.944591235003101921300981709370,
1.944591235003101921300981709370, 3.22773639167905402892302632100, 5.970300793360114507675288812948, 7.32744087186156568040361758365, 8.79080826384926706085041087801, 9.64611657141885916446348243181, 10.669447748894294158852750249409, 12.8133754336178107707525794242, 13.789131049820828862752138411306, 15.489382874162138740862518318787, 16.29315012627397787134601995636, 18.00576347603514123546512991036, 18.672598721958749196534732624781, 20.057441138757320313363140006188, 20.78392187781689933742302267641, 22.03419632198522342627092944428, 24.07370651255462611318217007938, 25.17795901381565984513361748684, 26.05503235862351310609443033328, 26.44569956073799372286794402195, 28.223749692000577024700664824476, 29.098555856325775051405658006562, 30.07959899261099683320680980451, 31.40500258932120590373403663653, 32.82573411853098146644267018675
