| L(s) = 1 | − 5-s + 13-s − 2·17-s − 4·19-s + 4·23-s + 25-s − 6·29-s − 8·31-s + 6·37-s + 2·41-s + 4·43-s − 7·49-s + 6·53-s + 2·61-s − 65-s + 8·67-s + 6·73-s + 4·79-s + 12·83-s + 2·85-s − 6·89-s + 4·95-s − 10·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.277·13-s − 0.485·17-s − 0.917·19-s + 0.834·23-s + 1/5·25-s − 1.11·29-s − 1.43·31-s + 0.986·37-s + 0.312·41-s + 0.609·43-s − 49-s + 0.824·53-s + 0.256·61-s − 0.124·65-s + 0.977·67-s + 0.702·73-s + 0.450·79-s + 1.31·83-s + 0.216·85-s − 0.635·89-s + 0.410·95-s − 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 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 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−15.13895659421010, −14.63440432138615, −14.25482081718137, −13.36441663175064, −13.02516481628957, −12.65522158214623, −11.98456179754290, −11.28317434528927, −10.96178582370890, −10.63440837508994, −9.647172981063850, −9.310551497952181, −8.708028256477821, −8.167302915190911, −7.570396057290924, −7.018849939167937, −6.475636264465154, −5.796311352230978, −5.207919526103153, −4.501205863652458, −3.907551180834241, −3.404339326172890, −2.492626092421087, −1.904442773298888, −0.9152762508617669, 0,
0.9152762508617669, 1.904442773298888, 2.492626092421087, 3.404339326172890, 3.907551180834241, 4.501205863652458, 5.207919526103153, 5.796311352230978, 6.475636264465154, 7.018849939167937, 7.570396057290924, 8.167302915190911, 8.708028256477821, 9.310551497952181, 9.647172981063850, 10.63440837508994, 10.96178582370890, 11.28317434528927, 11.98456179754290, 12.65522158214623, 13.02516481628957, 13.36441663175064, 14.25482081718137, 14.63440432138615, 15.13895659421010
