| L(s) = 1 | − 5-s − 11-s − 2·13-s − 2·17-s + 4·19-s + 25-s + 2·29-s − 2·37-s − 2·41-s + 12·43-s + 8·47-s − 7·49-s − 6·53-s + 55-s − 12·59-s + 6·61-s + 2·65-s − 4·67-s − 6·73-s + 16·79-s + 4·83-s + 2·85-s − 10·89-s − 4·95-s + 2·97-s − 6·101-s + 12·107-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.301·11-s − 0.554·13-s − 0.485·17-s + 0.917·19-s + 1/5·25-s + 0.371·29-s − 0.328·37-s − 0.312·41-s + 1.82·43-s + 1.16·47-s − 49-s − 0.824·53-s + 0.134·55-s − 1.56·59-s + 0.768·61-s + 0.248·65-s − 0.488·67-s − 0.702·73-s + 1.80·79-s + 0.439·83-s + 0.216·85-s − 1.05·89-s − 0.410·95-s + 0.203·97-s − 0.597·101-s + 1.16·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−7.61174917275683168608429949144, −6.88779650320825047145852837614, −6.14724620595704118269344592452, −5.31189242053979314165206064919, −4.69796496867100245349553385901, −3.93895813921455932627771770082, −3.06696550823289891463505972633, −2.35554930160436241743347894300, −1.18676833763413265366058848662, 0,
1.18676833763413265366058848662, 2.35554930160436241743347894300, 3.06696550823289891463505972633, 3.93895813921455932627771770082, 4.69796496867100245349553385901, 5.31189242053979314165206064919, 6.14724620595704118269344592452, 6.88779650320825047145852837614, 7.61174917275683168608429949144
