| L(s) = 1 | − 5-s − 11-s + 2·13-s − 6·17-s + 4·19-s + 4·23-s + 25-s − 6·29-s + 8·31-s − 2·37-s − 2·41-s − 4·43-s − 12·47-s − 7·49-s + 2·53-s + 55-s + 4·59-s − 10·61-s − 2·65-s + 16·67-s + 8·71-s + 14·73-s − 8·79-s − 4·83-s + 6·85-s − 10·89-s − 4·95-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.301·11-s + 0.554·13-s − 1.45·17-s + 0.917·19-s + 0.834·23-s + 1/5·25-s − 1.11·29-s + 1.43·31-s − 0.328·37-s − 0.312·41-s − 0.609·43-s − 1.75·47-s − 49-s + 0.274·53-s + 0.134·55-s + 0.520·59-s − 1.28·61-s − 0.248·65-s + 1.95·67-s + 0.949·71-s + 1.63·73-s − 0.900·79-s − 0.439·83-s + 0.650·85-s − 1.05·89-s − 0.410·95-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 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 16 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 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
−7.52567387860361416690294259657, −6.73114079621071100714169312036, −6.32256355952646480381816699905, −5.19072755325840146238482378089, −4.80763834609260757451310676100, −3.82573290551710488813917015410, −3.19776287627196162761105647639, −2.27052010539287522961646416611, −1.21814035480499692293997625850, 0,
1.21814035480499692293997625850, 2.27052010539287522961646416611, 3.19776287627196162761105647639, 3.82573290551710488813917015410, 4.80763834609260757451310676100, 5.19072755325840146238482378089, 6.32256355952646480381816699905, 6.73114079621071100714169312036, 7.52567387860361416690294259657
