| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
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
−12.32237201020811695553306360796, −11.17158952591131540748894685748, −10.08755192862193851674052925045, −9.026219897673102285381807836202, −7.77397120410345652122810585011, −6.58516209533226628464575787824, −5.23407411169190455619298184721, −3.76291998709216436997159381084, −2.07232651685692341885779650387, 0,
2.07232651685692341885779650387, 3.76291998709216436997159381084, 5.23407411169190455619298184721, 6.58516209533226628464575787824, 7.77397120410345652122810585011, 9.026219897673102285381807836202, 10.08755192862193851674052925045, 11.17158952591131540748894685748, 12.32237201020811695553306360796
