| L(s) = 1 | + 2·2-s + 3·3-s + 4·4-s + 4.39·5-s + 6·6-s + 7·7-s + 8·8-s + 9·9-s + 8.78·10-s + 50.1·11-s + 12·12-s + 11.4·13-s + 14·14-s + 13.1·15-s + 16·16-s + 107.·17-s + 18·18-s + 19·19-s + 17.5·20-s + 21·21-s + 100.·22-s − 181.·23-s + 24·24-s − 105.·25-s + 22.9·26-s + 27·27-s + 28·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.392·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 0.333·9-s + 0.277·10-s + 1.37·11-s + 0.288·12-s + 0.244·13-s + 0.267·14-s + 0.226·15-s + 0.250·16-s + 1.53·17-s + 0.235·18-s + 0.229·19-s + 0.196·20-s + 0.218·21-s + 0.971·22-s − 1.64·23-s + 0.204·24-s − 0.845·25-s + 0.173·26-s + 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.190188469\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.190188469\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 3 | \( 1 - 3T \) |
| 7 | \( 1 - 7T \) |
| 19 | \( 1 - 19T \) |
| good | 5 | \( 1 - 4.39T + 125T^{2} \) |
| 11 | \( 1 - 50.1T + 1.33e3T^{2} \) |
| 13 | \( 1 - 11.4T + 2.19e3T^{2} \) |
| 17 | \( 1 - 107.T + 4.91e3T^{2} \) |
| 23 | \( 1 + 181.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 286.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 85.4T + 2.97e4T^{2} \) |
| 37 | \( 1 - 313.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 18.8T + 6.89e4T^{2} \) |
| 43 | \( 1 - 395.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 364.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 152.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 375.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 253.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 291.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 587.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 788.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 929.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 371.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 914.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 374.T + 9.12e5T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.721626286440941666967277283521, −9.218843065289097000880638221309, −7.949716217421617354939913963149, −7.40360098163101451066690947611, −6.10513715377299416031223415152, −5.60079944768666754458588170464, −4.12527041213543073867454797320, −3.64643314807451000373069265319, −2.22758824285708646146733114899, −1.27935284882845820672065244683,
1.27935284882845820672065244683, 2.22758824285708646146733114899, 3.64643314807451000373069265319, 4.12527041213543073867454797320, 5.60079944768666754458588170464, 6.10513715377299416031223415152, 7.40360098163101451066690947611, 7.949716217421617354939913963149, 9.218843065289097000880638221309, 9.721626286440941666967277283521
