| L(s) = 1 | + 5·5-s − 7·7-s + 9·9-s − 6·17-s + 18·19-s + 25·25-s − 35·35-s + 66·37-s − 54·43-s + 45·45-s + 66·47-s + 49·49-s + 34·53-s − 62·59-s − 102·61-s − 63·63-s − 6·67-s + 138·71-s + 106·73-s + 122·79-s + 81·81-s − 30·85-s + 90·95-s − 166·97-s − 22·101-s − 46·103-s + 74·107-s + ⋯ |
| L(s) = 1 | + 5-s − 7-s + 9-s − 0.352·17-s + 0.947·19-s + 25-s − 35-s + 1.78·37-s − 1.25·43-s + 45-s + 1.40·47-s + 49-s + 0.641·53-s − 1.05·59-s − 1.67·61-s − 63-s − 0.0895·67-s + 1.94·71-s + 1.45·73-s + 1.54·79-s + 81-s − 0.352·85-s + 0.947·95-s − 1.71·97-s − 0.217·101-s − 0.446·103-s + 0.691·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.331563037\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.331563037\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - p T \) |
| 7 | \( 1 + p T \) |
| good | 3 | \( ( 1 - p T )( 1 + p T ) \) |
| 11 | \( ( 1 - p T )( 1 + p T ) \) |
| 13 | \( ( 1 - p T )( 1 + p T ) \) |
| 17 | \( 1 + 6 T + p^{2} T^{2} \) |
| 19 | \( 1 - 18 T + p^{2} T^{2} \) |
| 23 | \( ( 1 - p T )( 1 + p T ) \) |
| 29 | \( ( 1 - p T )( 1 + p T ) \) |
| 31 | \( ( 1 - p T )( 1 + p T ) \) |
| 37 | \( 1 - 66 T + p^{2} T^{2} \) |
| 41 | \( ( 1 - p T )( 1 + p T ) \) |
| 43 | \( 1 + 54 T + p^{2} T^{2} \) |
| 47 | \( 1 - 66 T + p^{2} T^{2} \) |
| 53 | \( 1 - 34 T + p^{2} T^{2} \) |
| 59 | \( 1 + 62 T + p^{2} T^{2} \) |
| 61 | \( 1 + 102 T + p^{2} T^{2} \) |
| 67 | \( 1 + 6 T + p^{2} T^{2} \) |
| 71 | \( 1 - 138 T + p^{2} T^{2} \) |
| 73 | \( 1 - 106 T + p^{2} T^{2} \) |
| 79 | \( 1 - 122 T + p^{2} T^{2} \) |
| 83 | \( ( 1 - p T )( 1 + p T ) \) |
| 89 | \( ( 1 - p T )( 1 + p T ) \) |
| 97 | \( 1 + 166 T + p^{2} T^{2} \) |
| 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
−9.539428180905766564035868246676, −9.237548580168835632250496072996, −7.917148230415813074585890404489, −6.96456117064053345440068062153, −6.34996038780446208763351959162, −5.47044453498617335291090912431, −4.43621617796695487071192053151, −3.30244356893919451877389955993, −2.23625062455508895166111645771, −0.962424687865905484867427142603,
0.962424687865905484867427142603, 2.23625062455508895166111645771, 3.30244356893919451877389955993, 4.43621617796695487071192053151, 5.47044453498617335291090912431, 6.34996038780446208763351959162, 6.96456117064053345440068062153, 7.917148230415813074585890404489, 9.237548580168835632250496072996, 9.539428180905766564035868246676
