| L(s) = 1 | − 4·2-s + 16·4-s + 6·5-s − 64·8-s − 24·10-s + 666·11-s − 559·13-s + 256·16-s + 1.74e3·17-s + 1.15e3·19-s + 96·20-s − 2.66e3·22-s + 3.46e3·23-s − 3.08e3·25-s + 2.23e3·26-s − 3.37e3·29-s + 6.29e3·31-s − 1.02e3·32-s − 6.96e3·34-s + 3.13e3·37-s − 4.62e3·38-s − 384·40-s + 4.86e3·41-s − 1.14e4·43-s + 1.06e4·44-s − 1.38e4·46-s − 2.31e3·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.107·5-s − 0.353·8-s − 0.0758·10-s + 1.65·11-s − 0.917·13-s + 1/4·16-s + 1.46·17-s + 0.735·19-s + 0.0536·20-s − 1.17·22-s + 1.36·23-s − 0.988·25-s + 0.648·26-s − 0.744·29-s + 1.17·31-s − 0.176·32-s − 1.03·34-s + 0.375·37-s − 0.519·38-s − 0.0379·40-s + 0.452·41-s − 0.940·43-s + 0.829·44-s − 0.966·46-s − 0.152·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.003741723\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.003741723\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p^{2} T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 6 T + p^{5} T^{2} \) |
| 11 | \( 1 - 666 T + p^{5} T^{2} \) |
| 13 | \( 1 + 43 p T + p^{5} T^{2} \) |
| 17 | \( 1 - 1740 T + p^{5} T^{2} \) |
| 19 | \( 1 - 1157 T + p^{5} T^{2} \) |
| 23 | \( 1 - 3468 T + p^{5} T^{2} \) |
| 29 | \( 1 + 3372 T + p^{5} T^{2} \) |
| 31 | \( 1 - 203 p T + p^{5} T^{2} \) |
| 37 | \( 1 - 3131 T + p^{5} T^{2} \) |
| 41 | \( 1 - 4866 T + p^{5} T^{2} \) |
| 43 | \( 1 + 11407 T + p^{5} T^{2} \) |
| 47 | \( 1 + 2310 T + p^{5} T^{2} \) |
| 53 | \( 1 - 28296 T + p^{5} T^{2} \) |
| 59 | \( 1 + 20544 T + p^{5} T^{2} \) |
| 61 | \( 1 + 4630 T + p^{5} T^{2} \) |
| 67 | \( 1 + 18745 T + p^{5} T^{2} \) |
| 71 | \( 1 - 38226 T + p^{5} T^{2} \) |
| 73 | \( 1 - 70589 T + p^{5} T^{2} \) |
| 79 | \( 1 + 62293 T + p^{5} T^{2} \) |
| 83 | \( 1 + 79818 T + p^{5} T^{2} \) |
| 89 | \( 1 - 18120 T + p^{5} T^{2} \) |
| 97 | \( 1 - 124754 T + p^{5} T^{2} \) |
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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
−9.664349138323203778659196575783, −8.658495115017593443378072223597, −7.66557689884245196807138029878, −7.02076879289246138666925747518, −6.08333780203388826245278855734, −5.11487495718468250589765372955, −3.85265685426776837937786103072, −2.87139104820102573209725856815, −1.55514932837920553532004312843, −0.76314077607542482855287364593,
0.76314077607542482855287364593, 1.55514932837920553532004312843, 2.87139104820102573209725856815, 3.85265685426776837937786103072, 5.11487495718468250589765372955, 6.08333780203388826245278855734, 7.02076879289246138666925747518, 7.66557689884245196807138029878, 8.658495115017593443378072223597, 9.664349138323203778659196575783
