| L(s) = 1 | − 128·4-s + 1.76e3·7-s + 1.26e4·13-s + 1.63e4·16-s + 1.43e4·19-s − 7.81e4·25-s − 2.25e5·28-s + 1.78e5·31-s − 6.15e5·37-s + 1.03e6·43-s + 2.28e6·49-s − 1.61e6·52-s + 1.53e6·61-s − 2.09e6·64-s − 4.05e6·67-s + 1.23e6·73-s − 1.83e6·76-s − 4.24e6·79-s + 2.22e7·91-s + 5.27e6·97-s + 1.00e7·100-s − 2.19e7·103-s − 1.68e7·109-s + 2.88e7·112-s + ⋯ |
| L(s) = 1 | − 4-s + 1.94·7-s + 1.59·13-s + 16-s + 0.480·19-s − 25-s − 1.94·28-s + 1.07·31-s − 1.99·37-s + 1.98·43-s + 2.77·49-s − 1.59·52-s + 0.867·61-s − 64-s − 1.64·67-s + 0.372·73-s − 0.480·76-s − 0.968·79-s + 3.09·91-s + 0.586·97-s + 100-s − 1.97·103-s − 1.24·109-s + 1.94·112-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.687593335\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.687593335\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + p^{7} T^{2} \) |
| 5 | \( 1 + p^{7} T^{2} \) |
| 7 | \( 1 - 1763 T + p^{7} T^{2} \) |
| 11 | \( 1 + p^{7} T^{2} \) |
| 13 | \( 1 - 12605 T + p^{7} T^{2} \) |
| 17 | \( 1 + p^{7} T^{2} \) |
| 19 | \( 1 - 14357 T + p^{7} T^{2} \) |
| 23 | \( 1 + p^{7} T^{2} \) |
| 29 | \( 1 + p^{7} T^{2} \) |
| 31 | \( 1 - 178916 T + p^{7} T^{2} \) |
| 37 | \( 1 + 615373 T + p^{7} T^{2} \) |
| 41 | \( 1 + p^{7} T^{2} \) |
| 43 | \( 1 - 1035224 T + p^{7} T^{2} \) |
| 47 | \( 1 + p^{7} T^{2} \) |
| 53 | \( 1 + p^{7} T^{2} \) |
| 59 | \( 1 + p^{7} T^{2} \) |
| 61 | \( 1 - 1537199 T + p^{7} T^{2} \) |
| 67 | \( 1 + 4058455 T + p^{7} T^{2} \) |
| 71 | \( 1 + p^{7} T^{2} \) |
| 73 | \( 1 - 1236809 T + p^{7} T^{2} \) |
| 79 | \( 1 + 4245427 T + p^{7} T^{2} \) |
| 83 | \( 1 + p^{7} T^{2} \) |
| 89 | \( 1 + p^{7} T^{2} \) |
| 97 | \( 1 - 5276357 T + p^{7} T^{2} \) |
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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
−15.62932849580074790159665340098, −14.26954823710391307104756796925, −13.58984236213973330622125177934, −11.85573740135451679835441965854, −10.63657410577492088131203225292, −8.840930843078410613417059310081, −7.928980319345615143303494471427, −5.48994012156018519478681771019, −4.12754930942058071487655765591, −1.28314704450956920770587051843,
1.28314704450956920770587051843, 4.12754930942058071487655765591, 5.48994012156018519478681771019, 7.928980319345615143303494471427, 8.840930843078410613417059310081, 10.63657410577492088131203225292, 11.85573740135451679835441965854, 13.58984236213973330622125177934, 14.26954823710391307104756796925, 15.62932849580074790159665340098
