| L(s) = 1 | + 9·3-s − 34·5-s + 49·7-s + 81·9-s + 340·11-s + 454·13-s − 306·15-s − 798·17-s − 892·19-s + 441·21-s + 3.19e3·23-s − 1.96e3·25-s + 729·27-s − 8.24e3·29-s + 2.49e3·31-s + 3.06e3·33-s − 1.66e3·35-s + 9.79e3·37-s + 4.08e3·39-s + 1.98e4·41-s + 1.72e4·43-s − 2.75e3·45-s − 8.92e3·47-s + 2.40e3·49-s − 7.18e3·51-s + 150·53-s − 1.15e4·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.608·5-s + 0.377·7-s + 1/3·9-s + 0.847·11-s + 0.745·13-s − 0.351·15-s − 0.669·17-s − 0.566·19-s + 0.218·21-s + 1.25·23-s − 0.630·25-s + 0.192·27-s − 1.81·29-s + 0.466·31-s + 0.489·33-s − 0.229·35-s + 1.17·37-s + 0.430·39-s + 1.84·41-s + 1.42·43-s − 0.202·45-s − 0.589·47-s + 1/7·49-s − 0.386·51-s + 0.00733·53-s − 0.515·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.608311326\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.608311326\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p^{2} T \) |
| 7 | \( 1 - p^{2} T \) |
| good | 5 | \( 1 + 34 T + p^{5} T^{2} \) |
| 11 | \( 1 - 340 T + p^{5} T^{2} \) |
| 13 | \( 1 - 454 T + p^{5} T^{2} \) |
| 17 | \( 1 + 798 T + p^{5} T^{2} \) |
| 19 | \( 1 + 892 T + p^{5} T^{2} \) |
| 23 | \( 1 - 3192 T + p^{5} T^{2} \) |
| 29 | \( 1 + 8242 T + p^{5} T^{2} \) |
| 31 | \( 1 - 2496 T + p^{5} T^{2} \) |
| 37 | \( 1 - 9798 T + p^{5} T^{2} \) |
| 41 | \( 1 - 19834 T + p^{5} T^{2} \) |
| 43 | \( 1 - 17236 T + p^{5} T^{2} \) |
| 47 | \( 1 + 8928 T + p^{5} T^{2} \) |
| 53 | \( 1 - 150 T + p^{5} T^{2} \) |
| 59 | \( 1 - 42396 T + p^{5} T^{2} \) |
| 61 | \( 1 - 14758 T + p^{5} T^{2} \) |
| 67 | \( 1 - 1676 T + p^{5} T^{2} \) |
| 71 | \( 1 + 14568 T + p^{5} T^{2} \) |
| 73 | \( 1 - 78378 T + p^{5} T^{2} \) |
| 79 | \( 1 - 2272 T + p^{5} T^{2} \) |
| 83 | \( 1 - 37764 T + p^{5} T^{2} \) |
| 89 | \( 1 + 117286 T + p^{5} T^{2} \) |
| 97 | \( 1 - 10002 T + p^{5} T^{2} \) |
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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
−11.05703355156774468292632628458, −9.539647215135225232145350182252, −8.852902043513638274273940261398, −7.948196697666891224299296664988, −7.04924647583409911325506229709, −5.91154036232606829397198566739, −4.39375374417510443571817010023, −3.69732301842623012294772726858, −2.25233064069868240762843230113, −0.894635478060079159301213642759,
0.894635478060079159301213642759, 2.25233064069868240762843230113, 3.69732301842623012294772726858, 4.39375374417510443571817010023, 5.91154036232606829397198566739, 7.04924647583409911325506229709, 7.948196697666891224299296664988, 8.852902043513638274273940261398, 9.539647215135225232145350182252, 11.05703355156774468292632628458
