L(s) = 1 | − 2-s + 4-s − 7-s − 8-s − 9-s + 14-s + 16-s + 18-s + 2·23-s − 25-s − 28-s − 32-s − 36-s − 2·46-s + 49-s + 50-s + 56-s + 63-s + 64-s − 2·71-s + 72-s − 2·79-s + 81-s + 2·92-s − 98-s − 100-s − 112-s + ⋯ |
L(s) = 1 | − 2-s + 4-s − 7-s − 8-s − 9-s + 14-s + 16-s + 18-s + 2·23-s − 25-s − 28-s − 32-s − 36-s − 2·46-s + 49-s + 50-s + 56-s + 63-s + 64-s − 2·71-s + 72-s − 2·79-s + 81-s + 2·92-s − 98-s − 100-s − 112-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3043447492\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3043447492\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 7 | \( 1 + T \) |
good | 3 | \( 1 + T^{2} \) |
| 5 | \( 1 + T^{2} \) |
| 11 | \( ( 1 - T )( 1 + T ) \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( ( 1 - T )( 1 + T ) \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( ( 1 - T )^{2} \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( ( 1 + T )^{2} \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( ( 1 + T )^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( ( 1 - T )( 1 + T ) \) |
| 97 | \( ( 1 - T )( 1 + T ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.79565908061006327148787841771, −14.73671678134309240431347460908, −13.20645456228466406130518818003, −11.90385973789881314134860691333, −10.82592427374314981434862061529, −9.573887196327588428321274714476, −8.622836713107787057397793052079, −7.13558216504220631858842167307, −5.85468181397150007828472600291, −3.00942554599034889633095542267,
3.00942554599034889633095542267, 5.85468181397150007828472600291, 7.13558216504220631858842167307, 8.622836713107787057397793052079, 9.573887196327588428321274714476, 10.82592427374314981434862061529, 11.90385973789881314134860691333, 13.20645456228466406130518818003, 14.73671678134309240431347460908, 15.79565908061006327148787841771