| L(s) = 1 | − 5-s + 2·13-s + 2·17-s + 4·19-s + 8·23-s + 25-s − 2·29-s − 8·31-s + 6·37-s + 2·41-s − 4·43-s − 7·49-s − 6·53-s + 4·59-s + 10·61-s − 2·65-s + 12·67-s − 2·73-s + 8·79-s − 12·83-s − 2·85-s + 6·89-s − 4·95-s − 14·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.554·13-s + 0.485·17-s + 0.917·19-s + 1.66·23-s + 1/5·25-s − 0.371·29-s − 1.43·31-s + 0.986·37-s + 0.312·41-s − 0.609·43-s − 49-s − 0.824·53-s + 0.520·59-s + 1.28·61-s − 0.248·65-s + 1.46·67-s − 0.234·73-s + 0.900·79-s − 1.31·83-s − 0.216·85-s + 0.635·89-s − 0.410·95-s − 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 87120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 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 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 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 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−14.37144579401055, −13.50667493812517, −13.18234213761137, −12.65499455971281, −12.30177830196170, −11.39385318445964, −11.29200942809512, −10.91642325752809, −10.09808702455975, −9.626736313214678, −9.162423220911494, −8.661683890467063, −7.965128875840438, −7.694886569438122, −6.944874174831483, −6.680240603878314, −5.851404906361481, −5.244851749900080, −5.000893206579399, −4.044703388558885, −3.642072332144656, −3.062571353179018, −2.451088858442685, −1.439361677003333, −1.000460577093279, 0,
1.000460577093279, 1.439361677003333, 2.451088858442685, 3.062571353179018, 3.642072332144656, 4.044703388558885, 5.000893206579399, 5.244851749900080, 5.851404906361481, 6.680240603878314, 6.944874174831483, 7.694886569438122, 7.965128875840438, 8.661683890467063, 9.162423220911494, 9.626736313214678, 10.09808702455975, 10.91642325752809, 11.29200942809512, 11.39385318445964, 12.30177830196170, 12.65499455971281, 13.18234213761137, 13.50667493812517, 14.37144579401055
