| L(s) = 1 | + 25·7-s − 427·13-s + 1.71e3·19-s − 3.12e3·25-s + 1.03e4·31-s − 6.66e3·37-s + 3.35e3·43-s − 1.61e4·49-s + 5.69e4·61-s + 3.79e4·67-s + 7.95e4·73-s − 9.08e4·79-s − 1.06e4·91-s + 1.77e5·97-s + 2.11e5·103-s + 1.14e5·109-s + ⋯ |
| L(s) = 1 | + 0.192·7-s − 0.700·13-s + 1.08·19-s − 25-s + 1.92·31-s − 0.799·37-s + 0.276·43-s − 0.962·49-s + 1.95·61-s + 1.03·67-s + 1.74·73-s − 1.63·79-s − 0.135·91-s + 1.91·97-s + 1.96·103-s + 0.922·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.022414158\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.022414158\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + p^{5} T^{2} \) |
| 7 | \( 1 - 25 T + p^{5} T^{2} \) |
| 11 | \( 1 + p^{5} T^{2} \) |
| 13 | \( 1 + 427 T + p^{5} T^{2} \) |
| 17 | \( 1 + p^{5} T^{2} \) |
| 19 | \( 1 - 1711 T + p^{5} T^{2} \) |
| 23 | \( 1 + p^{5} T^{2} \) |
| 29 | \( 1 + p^{5} T^{2} \) |
| 31 | \( 1 - 10324 T + p^{5} T^{2} \) |
| 37 | \( 1 + 6661 T + p^{5} T^{2} \) |
| 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 | \( 1 - 56927 T + p^{5} T^{2} \) |
| 67 | \( 1 - 37939 T + p^{5} T^{2} \) |
| 71 | \( 1 + p^{5} T^{2} \) |
| 73 | \( 1 - 79577 T + p^{5} T^{2} \) |
| 79 | \( 1 + 90857 T + p^{5} T^{2} \) |
| 83 | \( 1 + p^{5} T^{2} \) |
| 89 | \( 1 + p^{5} T^{2} \) |
| 97 | \( 1 - 177725 T + p^{5} T^{2} \) |
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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
−10.15998882876008572841153979707, −9.622789881909277591363343356680, −8.442657433354608457546793502945, −7.62688121491567011532073945175, −6.66143224586439710506397662854, −5.51735294864617744699030209224, −4.59667476841286367676668447726, −3.34339625275452751082977192509, −2.13335921013725123614775067971, −0.74051155464448931602971656666,
0.74051155464448931602971656666, 2.13335921013725123614775067971, 3.34339625275452751082977192509, 4.59667476841286367676668447726, 5.51735294864617744699030209224, 6.66143224586439710506397662854, 7.62688121491567011532073945175, 8.442657433354608457546793502945, 9.622789881909277591363343356680, 10.15998882876008572841153979707
