L(s) = 1 | + 2-s + 4-s − 5-s + 8-s − 10-s − 4·11-s + 13-s + 16-s − 4·19-s − 20-s − 4·22-s − 4·23-s + 25-s + 26-s − 6·29-s + 4·31-s + 32-s − 6·37-s − 4·38-s − 40-s − 2·41-s + 4·43-s − 4·44-s − 4·46-s + 8·47-s − 7·49-s + 50-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.353·8-s − 0.316·10-s − 1.20·11-s + 0.277·13-s + 1/4·16-s − 0.917·19-s − 0.223·20-s − 0.852·22-s − 0.834·23-s + 1/5·25-s + 0.196·26-s − 1.11·29-s + 0.718·31-s + 0.176·32-s − 0.986·37-s − 0.648·38-s − 0.158·40-s − 0.312·41-s + 0.609·43-s − 0.603·44-s − 0.589·46-s + 1.16·47-s − 49-s + 0.141·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 338130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338130 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3988528549\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3988528549\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 \) |
good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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
−12.50294785868600, −12.17003058281838, −11.89412639648836, −11.12044456994091, −10.82783178535317, −10.48901523731591, −10.05078901072603, −9.335997900733929, −8.921120565435811, −8.246693579879459, −7.975877816549868, −7.479876667983836, −7.060438734804667, −6.386671816963143, −6.002963827305070, −5.527265382270232, −5.000959960383299, −4.469210028604058, −4.071975087647620, −3.507783492030423, −2.949931132289842, −2.455062117528879, −1.864100224993456, −1.250142612014124, −0.1390213890911635,
0.1390213890911635, 1.250142612014124, 1.864100224993456, 2.455062117528879, 2.949931132289842, 3.507783492030423, 4.071975087647620, 4.469210028604058, 5.000959960383299, 5.527265382270232, 6.002963827305070, 6.386671816963143, 7.060438734804667, 7.479876667983836, 7.975877816549868, 8.246693579879459, 8.921120565435811, 9.335997900733929, 10.05078901072603, 10.48901523731591, 10.82783178535317, 11.12044456994091, 11.89412639648836, 12.17003058281838, 12.50294785868600