| L(s) = 1 | − 36·2-s − 81·3-s + 784·4-s − 1.31e3·5-s + 2.91e3·6-s − 4.48e3·7-s − 9.79e3·8-s + 6.56e3·9-s + 4.73e4·10-s + 1.47e3·11-s − 6.35e4·12-s − 1.51e5·13-s + 1.61e5·14-s + 1.06e5·15-s − 4.88e4·16-s + 1.08e5·17-s − 2.36e5·18-s + 5.93e5·19-s − 1.03e6·20-s + 3.62e5·21-s − 5.31e4·22-s − 9.69e5·23-s + 7.93e5·24-s − 2.26e5·25-s + 5.45e6·26-s − 5.31e5·27-s − 3.51e6·28-s + ⋯ |
| L(s) = 1 | − 1.59·2-s − 0.577·3-s + 1.53·4-s − 0.940·5-s + 0.918·6-s − 0.705·7-s − 0.845·8-s + 1/3·9-s + 1.49·10-s + 0.0303·11-s − 0.884·12-s − 1.47·13-s + 1.12·14-s + 0.542·15-s − 0.186·16-s + 0.314·17-s − 0.530·18-s + 1.04·19-s − 1.43·20-s + 0.407·21-s − 0.0483·22-s − 0.722·23-s + 0.487·24-s − 0.115·25-s + 2.34·26-s − 0.192·27-s − 1.07·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + p^{4} T \) |
| good | 2 | \( 1 + 9 p^{2} T + p^{9} T^{2} \) |
| 5 | \( 1 + 1314 T + p^{9} T^{2} \) |
| 7 | \( 1 + 640 p T + p^{9} T^{2} \) |
| 11 | \( 1 - 1476 T + p^{9} T^{2} \) |
| 13 | \( 1 + 151522 T + p^{9} T^{2} \) |
| 17 | \( 1 - 108162 T + p^{9} T^{2} \) |
| 19 | \( 1 - 593084 T + p^{9} T^{2} \) |
| 23 | \( 1 + 969480 T + p^{9} T^{2} \) |
| 29 | \( 1 + 6642522 T + p^{9} T^{2} \) |
| 31 | \( 1 - 7070600 T + p^{9} T^{2} \) |
| 37 | \( 1 + 7472410 T + p^{9} T^{2} \) |
| 41 | \( 1 + 4350150 T + p^{9} T^{2} \) |
| 43 | \( 1 + 4358716 T + p^{9} T^{2} \) |
| 47 | \( 1 - 28309248 T + p^{9} T^{2} \) |
| 53 | \( 1 - 16111710 T + p^{9} T^{2} \) |
| 59 | \( 1 + 86075964 T + p^{9} T^{2} \) |
| 61 | \( 1 - 32213918 T + p^{9} T^{2} \) |
| 67 | \( 1 - 99531452 T + p^{9} T^{2} \) |
| 71 | \( 1 + 44170488 T + p^{9} T^{2} \) |
| 73 | \( 1 + 23560630 T + p^{9} T^{2} \) |
| 79 | \( 1 + 401754760 T + p^{9} T^{2} \) |
| 83 | \( 1 + 744528708 T + p^{9} T^{2} \) |
| 89 | \( 1 - 769871034 T + p^{9} T^{2} \) |
| 97 | \( 1 - 907130882 T + p^{9} T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−24.45900041947759099865588309181, −22.54836108128621002407789390827, −19.94311036798915492992091017739, −18.82211231774437931318333889184, −17.12213500101042858454062767704, −15.81150029364889041545260315930, −11.83447952983302025015002870225, −9.848309622185260588522294441889, −7.47037563928868228952995554205, 0,
7.47037563928868228952995554205, 9.848309622185260588522294441889, 11.83447952983302025015002870225, 15.81150029364889041545260315930, 17.12213500101042858454062767704, 18.82211231774437931318333889184, 19.94311036798915492992091017739, 22.54836108128621002407789390827, 24.45900041947759099865588309181
