L(s) = 1 | − 2-s + 4-s − 5-s − 7-s − 8-s + 10-s + 11-s + 14-s + 16-s + 2·17-s + 19-s − 20-s − 22-s + 23-s − 28-s + 31-s − 32-s − 2·34-s + 35-s − 37-s − 38-s + 40-s − 41-s + 44-s − 46-s + 49-s − 55-s + ⋯ |
L(s) = 1 | − 2-s + 4-s − 5-s − 7-s − 8-s + 10-s + 11-s + 14-s + 16-s + 2·17-s + 19-s − 20-s − 22-s + 23-s − 28-s + 31-s − 32-s − 2·34-s + 35-s − 37-s − 38-s + 40-s − 41-s + 44-s − 46-s + 49-s − 55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5332209760\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5332209760\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 5 | \( 1 + T + T^{2} \) |
| 11 | \( 1 - T + T^{2} \) |
| 13 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( ( 1 - T )^{2} \) |
| 19 | \( 1 - T + T^{2} \) |
| 23 | \( 1 - T + T^{2} \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( 1 - T + T^{2} \) |
| 37 | \( 1 + T + T^{2} \) |
| 41 | \( 1 + T + T^{2} \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( 1 - T + T^{2} \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( 1 + T + T^{2} \) |
| 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
−10.28176671361445161923648708462, −9.709820991530204811583283397928, −8.906051884537861062525416013108, −7.972562304988798816726006679675, −7.24286208203389081154228430855, −6.50147871457115738114872337097, −5.39164668529538021141213699209, −3.66901223687432574881088411564, −3.09776787425856463816338690445, −1.10292190030766665047006066921,
1.10292190030766665047006066921, 3.09776787425856463816338690445, 3.66901223687432574881088411564, 5.39164668529538021141213699209, 6.50147871457115738114872337097, 7.24286208203389081154228430855, 7.972562304988798816726006679675, 8.906051884537861062525416013108, 9.709820991530204811583283397928, 10.28176671361445161923648708462