| L(s) = 1 | + 4·3-s + 12·5-s − 11·9-s − 12·11-s − 76·13-s + 48·15-s + 8·17-s − 100·19-s + 56·23-s + 19·25-s − 152·27-s − 166·29-s − 232·31-s − 48·33-s − 414·37-s − 304·39-s − 72·41-s + 452·43-s − 132·45-s + 424·47-s + 32·51-s − 18·53-s − 144·55-s − 400·57-s + 444·59-s + 284·61-s − 912·65-s + ⋯ |
| L(s) = 1 | + 0.769·3-s + 1.07·5-s − 0.407·9-s − 0.328·11-s − 1.62·13-s + 0.826·15-s + 0.114·17-s − 1.20·19-s + 0.507·23-s + 0.151·25-s − 1.08·27-s − 1.06·29-s − 1.34·31-s − 0.253·33-s − 1.83·37-s − 1.24·39-s − 0.274·41-s + 1.60·43-s − 0.437·45-s + 1.31·47-s + 0.0878·51-s − 0.0466·53-s − 0.353·55-s − 0.929·57-s + 0.979·59-s + 0.596·61-s − 1.74·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 4 T + p^{3} T^{2} \) |
| 5 | \( 1 - 12 T + p^{3} T^{2} \) |
| 11 | \( 1 + 12 T + p^{3} T^{2} \) |
| 13 | \( 1 + 76 T + p^{3} T^{2} \) |
| 17 | \( 1 - 8 T + p^{3} T^{2} \) |
| 19 | \( 1 + 100 T + p^{3} T^{2} \) |
| 23 | \( 1 - 56 T + p^{3} T^{2} \) |
| 29 | \( 1 + 166 T + p^{3} T^{2} \) |
| 31 | \( 1 + 232 T + p^{3} T^{2} \) |
| 37 | \( 1 + 414 T + p^{3} T^{2} \) |
| 41 | \( 1 + 72 T + p^{3} T^{2} \) |
| 43 | \( 1 - 452 T + p^{3} T^{2} \) |
| 47 | \( 1 - 424 T + p^{3} T^{2} \) |
| 53 | \( 1 + 18 T + p^{3} T^{2} \) |
| 59 | \( 1 - 444 T + p^{3} T^{2} \) |
| 61 | \( 1 - 284 T + p^{3} T^{2} \) |
| 67 | \( 1 + 524 T + p^{3} T^{2} \) |
| 71 | \( 1 - 1008 T + p^{3} T^{2} \) |
| 73 | \( 1 + 896 T + p^{3} T^{2} \) |
| 79 | \( 1 - 40 T + p^{3} T^{2} \) |
| 83 | \( 1 - 1388 T + p^{3} T^{2} \) |
| 89 | \( 1 + 448 T + p^{3} T^{2} \) |
| 97 | \( 1 - 824 T + p^{3} T^{2} \) |
| 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
−9.298549899376134741240630620560, −8.893221985846163812697540125958, −7.75074892450749011233530970691, −7.00710156765679836854062931680, −5.77778596285642488303754870320, −5.14489181079308437263160354342, −3.77877279942763758579479536283, −2.50785869249308927385336511534, −2.00426849430140933941621703249, 0,
2.00426849430140933941621703249, 2.50785869249308927385336511534, 3.77877279942763758579479536283, 5.14489181079308437263160354342, 5.77778596285642488303754870320, 7.00710156765679836854062931680, 7.75074892450749011233530970691, 8.893221985846163812697540125958, 9.298549899376134741240630620560
