| L(s) = 1 | + 2-s + 3-s + 4-s − 5-s + 6-s − 4·7-s + 8-s + 9-s − 10-s − 5·11-s + 12-s + 13-s − 4·14-s − 15-s + 16-s + 3·17-s + 18-s + 6·19-s − 20-s − 4·21-s − 5·22-s − 23-s + 24-s + 25-s + 26-s + 27-s − 4·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s − 1.51·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s − 1.50·11-s + 0.288·12-s + 0.277·13-s − 1.06·14-s − 0.258·15-s + 1/4·16-s + 0.727·17-s + 0.235·18-s + 1.37·19-s − 0.223·20-s − 0.872·21-s − 1.06·22-s − 0.208·23-s + 0.204·24-s + 1/5·25-s + 0.196·26-s + 0.192·27-s − 0.755·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 374790 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 374790 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.075423767\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.075423767\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 31 | \( 1 \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 12 T + p T^{2} \) |
| 79 | \( 1 + 5 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
| 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
−12.73057254782896, −12.05878861645841, −11.84315687615711, −11.03622756435756, −10.80749757893616, −10.11843616846917, −9.830403196513971, −9.433550125542763, −8.921774133384455, −8.177129810360195, −7.869593483383646, −7.382971847565381, −7.070320262505020, −6.487684257419619, −5.817454081859659, −5.514312518147341, −5.113855270508765, −4.344012685749904, −3.672797411300659, −3.566752705843287, −2.938509230544087, −2.564059983752163, −2.032214938435006, −0.9759177884029036, −0.5027050331424441,
0.5027050331424441, 0.9759177884029036, 2.032214938435006, 2.564059983752163, 2.938509230544087, 3.566752705843287, 3.672797411300659, 4.344012685749904, 5.113855270508765, 5.514312518147341, 5.817454081859659, 6.487684257419619, 7.070320262505020, 7.382971847565381, 7.869593483383646, 8.177129810360195, 8.921774133384455, 9.433550125542763, 9.830403196513971, 10.11843616846917, 10.80749757893616, 11.03622756435756, 11.84315687615711, 12.05878861645841, 12.73057254782896
