L(s) = 1 | + 3-s + 4·7-s + 9-s + 4·13-s + 6·17-s + 19-s + 4·21-s + 6·23-s − 5·25-s + 27-s − 6·29-s − 2·31-s + 4·37-s + 4·39-s + 6·41-s − 4·43-s − 6·47-s + 9·49-s + 6·51-s − 6·53-s + 57-s − 12·59-s − 14·61-s + 4·63-s + 8·67-s + 6·69-s + 14·73-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 1.51·7-s + 1/3·9-s + 1.10·13-s + 1.45·17-s + 0.229·19-s + 0.872·21-s + 1.25·23-s − 25-s + 0.192·27-s − 1.11·29-s − 0.359·31-s + 0.657·37-s + 0.640·39-s + 0.937·41-s − 0.609·43-s − 0.875·47-s + 9/7·49-s + 0.840·51-s − 0.824·53-s + 0.132·57-s − 1.56·59-s − 1.79·61-s + 0.503·63-s + 0.977·67-s + 0.722·69-s + 1.63·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.273141381\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.273141381\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 19 | \( 1 - T \) |
good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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
−8.380140351181000236453607755385, −7.83146723424270026458224882246, −7.43283795707002150488374868194, −6.23636375738531983020866481834, −5.45716303902372547637431694432, −4.74514961136841231763588047323, −3.81199005087099183295491504054, −3.09156700676949380226753412779, −1.82297840574153350889823289813, −1.18152128549559051518186862695,
1.18152128549559051518186862695, 1.82297840574153350889823289813, 3.09156700676949380226753412779, 3.81199005087099183295491504054, 4.74514961136841231763588047323, 5.45716303902372547637431694432, 6.23636375738531983020866481834, 7.43283795707002150488374868194, 7.83146723424270026458224882246, 8.380140351181000236453607755385