L(s) = 1 | + 3-s + 5-s + 7-s − 11-s − 13-s + 15-s + 17-s + 21-s + 25-s − 27-s + 29-s − 33-s + 35-s − 39-s − 47-s + 49-s + 51-s − 55-s − 65-s − 2·71-s − 2·73-s + 75-s − 77-s + 79-s − 81-s − 2·83-s + 85-s + ⋯ |
L(s) = 1 | + 3-s + 5-s + 7-s − 11-s − 13-s + 15-s + 17-s + 21-s + 25-s − 27-s + 29-s − 33-s + 35-s − 39-s − 47-s + 49-s + 51-s − 55-s − 65-s − 2·71-s − 2·73-s + 75-s − 77-s + 79-s − 81-s − 2·83-s + 85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.958626298\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.958626298\) |
\(L(1)\) |
|
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 | \( 1 - T \) |
good | 3 | \( 1 - T + T^{2} \) |
| 11 | \( 1 + T + T^{2} \) |
| 13 | \( 1 + T + T^{2} \) |
| 17 | \( 1 - T + T^{2} \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( 1 - T + T^{2} \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( 1 + T + T^{2} \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( ( 1 + T )^{2} \) |
| 73 | \( ( 1 + T )^{2} \) |
| 79 | \( 1 - T + T^{2} \) |
| 83 | \( ( 1 + T )^{2} \) |
| 89 | \( ( 1 - T )( 1 + T ) \) |
| 97 | \( 1 - T + T^{2} \) |
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
−9.124509353505063248333293657547, −8.437391199479060024941774546356, −7.81843726316528054009667193924, −7.17044445879043346687486974805, −5.91316605065441977695989779646, −5.24734081919091668582870628636, −4.52378910882817524591378354822, −3.07890357616003525241526909737, −2.51807976290935731715877354947, −1.57296568307498081096017582090,
1.57296568307498081096017582090, 2.51807976290935731715877354947, 3.07890357616003525241526909737, 4.52378910882817524591378354822, 5.24734081919091668582870628636, 5.91316605065441977695989779646, 7.17044445879043346687486974805, 7.81843726316528054009667193924, 8.437391199479060024941774546356, 9.124509353505063248333293657547