| L(s) = 1 | − 3-s + 1.73·7-s − 2·9-s − 5.19·11-s − 3·17-s + 5.19·19-s − 1.73·21-s − 3·23-s − 5·25-s + 5·27-s − 9·29-s + 3.46·31-s + 5.19·33-s − 5.19·37-s + 5.19·41-s − 5·43-s − 10.3·47-s − 4·49-s + 3·51-s − 6·53-s − 5.19·57-s − 5.19·59-s − 5·61-s − 3.46·63-s − 1.73·67-s + 3·69-s + 5.19·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.654·7-s − 0.666·9-s − 1.56·11-s − 0.727·17-s + 1.19·19-s − 0.377·21-s − 0.625·23-s − 25-s + 0.962·27-s − 1.67·29-s + 0.622·31-s + 0.904·33-s − 0.854·37-s + 0.811·41-s − 0.762·43-s − 1.51·47-s − 0.571·49-s + 0.420·51-s − 0.824·53-s − 0.688·57-s − 0.676·59-s − 0.640·61-s − 0.436·63-s − 0.211·67-s + 0.361·69-s + 0.616·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 676 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 676 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T + 3T^{2} \) |
| 5 | \( 1 + 5T^{2} \) |
| 7 | \( 1 - 1.73T + 7T^{2} \) |
| 11 | \( 1 + 5.19T + 11T^{2} \) |
| 17 | \( 1 + 3T + 17T^{2} \) |
| 19 | \( 1 - 5.19T + 19T^{2} \) |
| 23 | \( 1 + 3T + 23T^{2} \) |
| 29 | \( 1 + 9T + 29T^{2} \) |
| 31 | \( 1 - 3.46T + 31T^{2} \) |
| 37 | \( 1 + 5.19T + 37T^{2} \) |
| 41 | \( 1 - 5.19T + 41T^{2} \) |
| 43 | \( 1 + 5T + 43T^{2} \) |
| 47 | \( 1 + 10.3T + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 + 5.19T + 59T^{2} \) |
| 61 | \( 1 + 5T + 61T^{2} \) |
| 67 | \( 1 + 1.73T + 67T^{2} \) |
| 71 | \( 1 - 5.19T + 71T^{2} \) |
| 73 | \( 1 - 6.92T + 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 - 10.3T + 83T^{2} \) |
| 89 | \( 1 - 15.5T + 89T^{2} \) |
| 97 | \( 1 + 12.1T + 97T^{2} \) |
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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
−10.20318100276792772256039355824, −9.267372578112334890932185058614, −8.086526968598507642942107586040, −7.66774610197130324771576277633, −6.32293638585806870573383950209, −5.41026755876775665389390402985, −4.83436974752246048513535758180, −3.31480386211908552493741942260, −2.02690570501970823338774114044, 0,
2.02690570501970823338774114044, 3.31480386211908552493741942260, 4.83436974752246048513535758180, 5.41026755876775665389390402985, 6.32293638585806870573383950209, 7.66774610197130324771576277633, 8.086526968598507642942107586040, 9.267372578112334890932185058614, 10.20318100276792772256039355824
