| L(s) = 1 | + 3-s + 5-s + 9-s + 2·13-s + 15-s − 6·17-s − 4·19-s + 4·23-s + 25-s + 27-s − 2·29-s + 2·37-s + 2·39-s − 6·41-s − 4·43-s + 45-s − 7·49-s − 6·51-s + 10·53-s − 4·57-s − 12·59-s − 10·61-s + 2·65-s − 4·67-s + 4·69-s − 14·73-s + 75-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 1/3·9-s + 0.554·13-s + 0.258·15-s − 1.45·17-s − 0.917·19-s + 0.834·23-s + 1/5·25-s + 0.192·27-s − 0.371·29-s + 0.328·37-s + 0.320·39-s − 0.937·41-s − 0.609·43-s + 0.149·45-s − 49-s − 0.840·51-s + 1.37·53-s − 0.529·57-s − 1.56·59-s − 1.28·61-s + 0.248·65-s − 0.488·67-s + 0.481·69-s − 1.63·73-s + 0.115·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.002749876\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.002749876\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 31 | \( 1 \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−13.09752437405142, −12.62893076951266, −12.01129608951080, −11.49423874325870, −10.94134189840104, −10.66677794215777, −10.14415488628096, −9.590615971786105, −9.037429471410643, −8.747819763687371, −8.440111707716698, −7.705053513664234, −7.270695593572524, −6.613524040557751, −6.375757036912737, −5.819980301208583, −5.077188437449462, −4.627214233456157, −4.155478538310278, −3.548761384323406, −2.901941361194672, −2.496769323226594, −1.699027075610869, −1.467211785566502, −0.3509013434806924,
0.3509013434806924, 1.467211785566502, 1.699027075610869, 2.496769323226594, 2.901941361194672, 3.548761384323406, 4.155478538310278, 4.627214233456157, 5.077188437449462, 5.819980301208583, 6.375757036912737, 6.613524040557751, 7.270695593572524, 7.705053513664234, 8.440111707716698, 8.747819763687371, 9.037429471410643, 9.590615971786105, 10.14415488628096, 10.66677794215777, 10.94134189840104, 11.49423874325870, 12.01129608951080, 12.62893076951266, 13.09752437405142
