L(s) = 1 | + 5-s − 9-s + 25-s + 2·29-s + 2·41-s − 45-s + 2·61-s + 81-s − 2·89-s − 2·101-s − 2·109-s + ⋯ |
L(s) = 1 | + 5-s − 9-s + 25-s + 2·29-s + 2·41-s − 45-s + 2·61-s + 81-s − 2·89-s − 2·101-s − 2·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.495871664\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.495871664\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + T^{2} \) |
| 11 | \( ( 1 - T )( 1 + T ) \) |
| 13 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( ( 1 - T )( 1 + T ) \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( ( 1 - T )^{2} \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( ( 1 - T )^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( ( 1 - T )^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( ( 1 + T )^{2} \) |
| 97 | \( ( 1 - T )( 1 + T ) \) |
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
−8.615116509690873015394710039139, −8.122884554386866985375789998908, −7.03335988148168737788661406501, −6.35236334832955575736344900573, −5.70560789279347720667136474143, −5.06080327143520332678135689513, −4.10976657797257055296249118587, −2.89613286929450524344979657331, −2.40654091547937943441348225391, −1.08767216799225863295325335736,
1.08767216799225863295325335736, 2.40654091547937943441348225391, 2.89613286929450524344979657331, 4.10976657797257055296249118587, 5.06080327143520332678135689513, 5.70560789279347720667136474143, 6.35236334832955575736344900573, 7.03335988148168737788661406501, 8.122884554386866985375789998908, 8.615116509690873015394710039139