| L(s) = 1 | + 4·3-s + 2·5-s + 24·7-s − 11·9-s + 44·11-s − 22·13-s + 8·15-s + 50·17-s − 44·19-s + 96·21-s − 56·23-s − 121·25-s − 152·27-s − 198·29-s − 160·31-s + 176·33-s + 48·35-s + 162·37-s − 88·39-s − 198·41-s − 52·43-s − 22·45-s + 528·47-s + 233·49-s + 200·51-s + 242·53-s + 88·55-s + ⋯ |
| L(s) = 1 | + 0.769·3-s + 0.178·5-s + 1.29·7-s − 0.407·9-s + 1.20·11-s − 0.469·13-s + 0.137·15-s + 0.713·17-s − 0.531·19-s + 0.997·21-s − 0.507·23-s − 0.967·25-s − 1.08·27-s − 1.26·29-s − 0.926·31-s + 0.928·33-s + 0.231·35-s + 0.719·37-s − 0.361·39-s − 0.754·41-s − 0.184·43-s − 0.0728·45-s + 1.63·47-s + 0.679·49-s + 0.549·51-s + 0.627·53-s + 0.215·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.951702858\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.951702858\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 - 4 T + p^{3} T^{2} \) |
| 5 | \( 1 - 2 T + p^{3} T^{2} \) |
| 7 | \( 1 - 24 T + p^{3} T^{2} \) |
| 11 | \( 1 - 4 p T + p^{3} T^{2} \) |
| 13 | \( 1 + 22 T + p^{3} T^{2} \) |
| 17 | \( 1 - 50 T + p^{3} T^{2} \) |
| 19 | \( 1 + 44 T + p^{3} T^{2} \) |
| 23 | \( 1 + 56 T + p^{3} T^{2} \) |
| 29 | \( 1 + 198 T + p^{3} T^{2} \) |
| 31 | \( 1 + 160 T + p^{3} T^{2} \) |
| 37 | \( 1 - 162 T + p^{3} T^{2} \) |
| 41 | \( 1 + 198 T + p^{3} T^{2} \) |
| 43 | \( 1 + 52 T + p^{3} T^{2} \) |
| 47 | \( 1 - 528 T + p^{3} T^{2} \) |
| 53 | \( 1 - 242 T + p^{3} T^{2} \) |
| 59 | \( 1 - 668 T + p^{3} T^{2} \) |
| 61 | \( 1 + 550 T + p^{3} T^{2} \) |
| 67 | \( 1 + 188 T + p^{3} T^{2} \) |
| 71 | \( 1 - 728 T + p^{3} T^{2} \) |
| 73 | \( 1 - 154 T + p^{3} T^{2} \) |
| 79 | \( 1 + 656 T + p^{3} T^{2} \) |
| 83 | \( 1 + 236 T + p^{3} T^{2} \) |
| 89 | \( 1 - 714 T + p^{3} T^{2} \) |
| 97 | \( 1 + 478 T + p^{3} T^{2} \) |
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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
−14.52852129927464806220229296898, −13.66860383855412140109241011981, −12.08024806202359154497036910688, −11.16603412380226034776101475658, −9.593654230661643458274775907077, −8.539948533393353318133592215836, −7.49986312834430446464759915556, −5.65648094710408960300186932744, −3.93704783995578753079150211533, −1.94590534461252243639779566965,
1.94590534461252243639779566965, 3.93704783995578753079150211533, 5.65648094710408960300186932744, 7.49986312834430446464759915556, 8.539948533393353318133592215836, 9.593654230661643458274775907077, 11.16603412380226034776101475658, 12.08024806202359154497036910688, 13.66860383855412140109241011981, 14.52852129927464806220229296898
