| L(s) = 1 | − 32·4-s + 211·7-s + 775·13-s + 1.02e3·16-s + 3.14e3·19-s − 6.75e3·28-s − 1.03e4·31-s + 9.88e3·37-s + 3.35e3·43-s + 2.77e4·49-s − 2.48e4·52-s − 1.83e4·61-s − 3.27e4·64-s − 7.34e4·67-s + 7.81e4·73-s − 1.00e5·76-s + 9.70e3·79-s + 1.63e5·91-s + 4.33e4·97-s − 7.05e4·103-s + 1.14e5·109-s + 2.16e5·112-s + ⋯ |
| L(s) = 1 | − 4-s + 1.62·7-s + 1.27·13-s + 16-s + 1.99·19-s − 1.62·28-s − 1.92·31-s + 1.18·37-s + 0.276·43-s + 1.64·49-s − 1.27·52-s − 0.629·61-s − 64-s − 1.99·67-s + 1.71·73-s − 1.99·76-s + 0.174·79-s + 2.07·91-s + 0.467·97-s − 0.655·103-s + 0.922·109-s + 1.62·112-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.526382031\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.526382031\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + p^{5} T^{2} \) |
| 7 | \( 1 - 211 T + p^{5} T^{2} \) |
| 11 | \( 1 + p^{5} T^{2} \) |
| 13 | \( 1 - 775 T + p^{5} T^{2} \) |
| 17 | \( 1 + p^{5} T^{2} \) |
| 19 | \( 1 - 3143 T + p^{5} T^{2} \) |
| 23 | \( 1 + p^{5} T^{2} \) |
| 29 | \( 1 + p^{5} T^{2} \) |
| 31 | \( 1 + 10324 T + p^{5} T^{2} \) |
| 37 | \( 1 - 9889 T + p^{5} T^{2} \) |
| 41 | \( 1 + p^{5} T^{2} \) |
| 43 | \( 1 - 3352 T + p^{5} T^{2} \) |
| 47 | \( 1 + p^{5} T^{2} \) |
| 53 | \( 1 + p^{5} T^{2} \) |
| 59 | \( 1 + p^{5} T^{2} \) |
| 61 | \( 1 + 18301 T + p^{5} T^{2} \) |
| 67 | \( 1 + 73475 T + p^{5} T^{2} \) |
| 71 | \( 1 + p^{5} T^{2} \) |
| 73 | \( 1 - 78127 T + p^{5} T^{2} \) |
| 79 | \( 1 - 9707 T + p^{5} T^{2} \) |
| 83 | \( 1 + p^{5} T^{2} \) |
| 89 | \( 1 + p^{5} T^{2} \) |
| 97 | \( 1 - 43339 T + p^{5} T^{2} \) |
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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
−9.520773713349287032271136049640, −8.881396802079777796438591418745, −8.013198245550095855575685672725, −7.44748998423029770708490880021, −5.82039410681858797847324396626, −5.18536639224379646616011677590, −4.27924496387136233267262942248, −3.33344889319433605479580018620, −1.62570464003451532705992557966, −0.843097843074319443764515977568,
0.843097843074319443764515977568, 1.62570464003451532705992557966, 3.33344889319433605479580018620, 4.27924496387136233267262942248, 5.18536639224379646616011677590, 5.82039410681858797847324396626, 7.44748998423029770708490880021, 8.013198245550095855575685672725, 8.881396802079777796438591418745, 9.520773713349287032271136049640
