| L(s) = 1 | − 3-s − 5-s + 9-s − 2·13-s + 15-s − 6·17-s − 4·19-s − 8·23-s + 25-s − 27-s − 2·29-s + 4·31-s + 10·37-s + 2·39-s − 2·41-s + 4·43-s − 45-s + 8·47-s + 6·51-s − 2·53-s + 4·57-s + 8·59-s + 2·61-s + 2·65-s + 12·67-s + 8·69-s − 8·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1/3·9-s − 0.554·13-s + 0.258·15-s − 1.45·17-s − 0.917·19-s − 1.66·23-s + 1/5·25-s − 0.192·27-s − 0.371·29-s + 0.718·31-s + 1.64·37-s + 0.320·39-s − 0.312·41-s + 0.609·43-s − 0.149·45-s + 1.16·47-s + 0.840·51-s − 0.274·53-s + 0.529·57-s + 1.04·59-s + 0.256·61-s + 0.248·65-s + 1.46·67-s + 0.963·69-s − 0.949·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 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
−15.80785108872954, −15.23107821662168, −14.74511518562114, −14.14365887269232, −13.42480702189859, −13.03020954044254, −12.32910875105604, −11.98545937825038, −11.32788752298935, −10.90137437322746, −10.35002341261174, −9.687478569339392, −9.179195216642414, −8.369394832582831, −7.964723574342801, −7.276352098794111, −6.570140726441420, −6.214719550866048, −5.471946640824109, −4.659628160490358, −4.231756426348396, −3.687230850750082, −2.407657085985740, −2.157134661316001, −0.8207562090201125, 0,
0.8207562090201125, 2.157134661316001, 2.407657085985740, 3.687230850750082, 4.231756426348396, 4.659628160490358, 5.471946640824109, 6.214719550866048, 6.570140726441420, 7.276352098794111, 7.964723574342801, 8.369394832582831, 9.179195216642414, 9.687478569339392, 10.35002341261174, 10.90137437322746, 11.32788752298935, 11.98545937825038, 12.32910875105604, 13.03020954044254, 13.42480702189859, 14.14365887269232, 14.74511518562114, 15.23107821662168, 15.80785108872954
