| L(s) = 1 | + 3·3-s + 13·7-s + 9·9-s + 23·13-s − 11·19-s + 39·21-s + 27·27-s − 59·31-s + 26·37-s + 69·39-s − 83·43-s + 120·49-s − 33·57-s − 121·61-s + 117·63-s + 13·67-s − 46·73-s + 142·79-s + 81·81-s + 299·91-s − 177·93-s + 167·97-s − 194·103-s + 71·109-s + 78·111-s + 207·117-s + ⋯ |
| L(s) = 1 | + 3-s + 13/7·7-s + 9-s + 1.76·13-s − 0.578·19-s + 13/7·21-s + 27-s − 1.90·31-s + 0.702·37-s + 1.76·39-s − 1.93·43-s + 2.44·49-s − 0.578·57-s − 1.98·61-s + 13/7·63-s + 0.194·67-s − 0.630·73-s + 1.79·79-s + 81-s + 23/7·91-s − 1.90·93-s + 1.72·97-s − 1.88·103-s + 0.651·109-s + 0.702·111-s + 1.76·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.905636023\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.905636023\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 13 T + p^{2} T^{2} \) |
| 11 | \( ( 1 - p T )( 1 + p T ) \) |
| 13 | \( 1 - 23 T + p^{2} T^{2} \) |
| 17 | \( ( 1 - p T )( 1 + p T ) \) |
| 19 | \( 1 + 11 T + p^{2} T^{2} \) |
| 23 | \( ( 1 - p T )( 1 + p T ) \) |
| 29 | \( ( 1 - p T )( 1 + p T ) \) |
| 31 | \( 1 + 59 T + p^{2} T^{2} \) |
| 37 | \( 1 - 26 T + p^{2} T^{2} \) |
| 41 | \( ( 1 - p T )( 1 + p T ) \) |
| 43 | \( 1 + 83 T + p^{2} T^{2} \) |
| 47 | \( ( 1 - p T )( 1 + p T ) \) |
| 53 | \( ( 1 - p T )( 1 + p T ) \) |
| 59 | \( ( 1 - p T )( 1 + p T ) \) |
| 61 | \( 1 + 121 T + p^{2} T^{2} \) |
| 67 | \( 1 - 13 T + p^{2} T^{2} \) |
| 71 | \( ( 1 - p T )( 1 + p T ) \) |
| 73 | \( 1 + 46 T + p^{2} T^{2} \) |
| 79 | \( 1 - 142 T + p^{2} T^{2} \) |
| 83 | \( ( 1 - p T )( 1 + p T ) \) |
| 89 | \( ( 1 - p T )( 1 + p T ) \) |
| 97 | \( 1 - 167 T + p^{2} T^{2} \) |
| 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
−9.253688323529239818792069687481, −8.609382516400515872176794955426, −8.096974473625739139348438709692, −7.38811747714884730896397418171, −6.25741779837911633765282118398, −5.13674837859668960895420271998, −4.24518312535289265251678935599, −3.43427309688605128880713785429, −1.99906543379683976856385334014, −1.34377828702736940029089054832,
1.34377828702736940029089054832, 1.99906543379683976856385334014, 3.43427309688605128880713785429, 4.24518312535289265251678935599, 5.13674837859668960895420271998, 6.25741779837911633765282118398, 7.38811747714884730896397418171, 8.096974473625739139348438709692, 8.609382516400515872176794955426, 9.253688323529239818792069687481
