| L(s) = 1 | − 3-s − 5-s − 5·7-s − 2·9-s − 2·11-s + 15-s − 3·17-s − 2·19-s + 5·21-s + 4·23-s − 4·25-s + 5·27-s + 6·29-s + 4·31-s + 2·33-s + 5·35-s + 11·37-s − 8·41-s + 43-s + 2·45-s − 9·47-s + 18·49-s + 3·51-s + 12·53-s + 2·55-s + 2·57-s + 6·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 1.88·7-s − 2/3·9-s − 0.603·11-s + 0.258·15-s − 0.727·17-s − 0.458·19-s + 1.09·21-s + 0.834·23-s − 4/5·25-s + 0.962·27-s + 1.11·29-s + 0.718·31-s + 0.348·33-s + 0.845·35-s + 1.80·37-s − 1.24·41-s + 0.152·43-s + 0.298·45-s − 1.31·47-s + 18/7·49-s + 0.420·51-s + 1.64·53-s + 0.269·55-s + 0.264·57-s + 0.781·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10816 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10816 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + 5 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 11 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 6 T + p T^{2} \) |
| 71 | \( 1 + 7 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−16.75705290861548, −16.18439856110719, −15.84306116472247, −15.23358747728945, −14.70775155133387, −13.71714635078638, −13.21763970782725, −12.93821973685383, −12.06560042835683, −11.75845164409479, −11.00585020714639, −10.39539645552840, −9.884329890643130, −9.236776112045240, −8.520496358259161, −7.993441319581547, −6.939011241448557, −6.594802234782285, −6.027003633419859, −5.328600879948662, −4.505743288479360, −3.722567847104299, −2.912294681220556, −2.470164438967357, −0.7516906279247953, 0,
0.7516906279247953, 2.470164438967357, 2.912294681220556, 3.722567847104299, 4.505743288479360, 5.328600879948662, 6.027003633419859, 6.594802234782285, 6.939011241448557, 7.993441319581547, 8.520496358259161, 9.236776112045240, 9.884329890643130, 10.39539645552840, 11.00585020714639, 11.75845164409479, 12.06560042835683, 12.93821973685383, 13.21763970782725, 13.71714635078638, 14.70775155133387, 15.23358747728945, 15.84306116472247, 16.18439856110719, 16.75705290861548
