L(s) = 1 | + 3-s − 2·5-s + 7-s + 9-s + 2·13-s − 2·15-s − 17-s − 8·19-s + 21-s − 25-s + 27-s − 2·29-s − 2·35-s − 2·37-s + 2·39-s + 6·41-s + 4·43-s − 2·45-s + 8·47-s + 49-s − 51-s + 6·53-s − 8·57-s + 4·59-s − 6·61-s + 63-s − 4·65-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.894·5-s + 0.377·7-s + 1/3·9-s + 0.554·13-s − 0.516·15-s − 0.242·17-s − 1.83·19-s + 0.218·21-s − 1/5·25-s + 0.192·27-s − 0.371·29-s − 0.338·35-s − 0.328·37-s + 0.320·39-s + 0.937·41-s + 0.609·43-s − 0.298·45-s + 1.16·47-s + 1/7·49-s − 0.140·51-s + 0.824·53-s − 1.05·57-s + 0.520·59-s − 0.768·61-s + 0.125·63-s − 0.496·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22848 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22848 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 + T \) |
good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−15.65060205295933, −15.20417835490988, −14.80877834550554, −14.12208163702170, −13.73793920155757, −12.89563691962714, −12.67080559544949, −11.95527847253122, −11.34275973763140, −10.83884933465334, −10.43879630730763, −9.590638101025444, −8.919339434812601, −8.523176834920947, −7.998617641008728, −7.472217669243700, −6.824799358309610, −6.171795688505034, −5.455330187496640, −4.564561944519966, −3.982562565241969, −3.730735060682279, −2.602538065229727, −2.101742578405053, −1.076890530604203, 0,
1.076890530604203, 2.101742578405053, 2.602538065229727, 3.730735060682279, 3.982562565241969, 4.564561944519966, 5.455330187496640, 6.171795688505034, 6.824799358309610, 7.472217669243700, 7.998617641008728, 8.523176834920947, 8.919339434812601, 9.590638101025444, 10.43879630730763, 10.83884933465334, 11.34275973763140, 11.95527847253122, 12.67080559544949, 12.89563691962714, 13.73793920155757, 14.12208163702170, 14.80877834550554, 15.20417835490988, 15.65060205295933