| L(s) = 1 | − 5-s + 2·17-s + 2·19-s + 6·23-s + 25-s + 2·29-s − 6·37-s + 2·41-s − 2·43-s − 2·47-s − 7·49-s − 2·53-s + 8·59-s − 4·61-s + 4·67-s − 2·71-s + 10·73-s + 4·79-s − 12·83-s − 2·85-s − 2·95-s − 18·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.485·17-s + 0.458·19-s + 1.25·23-s + 1/5·25-s + 0.371·29-s − 0.986·37-s + 0.312·41-s − 0.304·43-s − 0.291·47-s − 49-s − 0.274·53-s + 1.04·59-s − 0.512·61-s + 0.488·67-s − 0.237·71-s + 1.17·73-s + 0.450·79-s − 1.31·83-s − 0.216·85-s − 0.205·95-s − 1.82·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-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 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−14.11678616051947, −13.72678963514116, −13.08657335456314, −12.65532903764547, −12.20118618857810, −11.66856904005622, −11.18407198350395, −10.78033232567414, −10.16325420990684, −9.632349046284574, −9.206746214918916, −8.491719587937563, −8.191347038913660, −7.555093606625250, −6.980378823231359, −6.668479466048576, −5.884356157093315, −5.262773659159644, −4.890365431657997, −4.207737267941194, −3.503208349644700, −3.111765051234592, −2.422170530367224, −1.520117029609025, −0.9273012256259188, 0,
0.9273012256259188, 1.520117029609025, 2.422170530367224, 3.111765051234592, 3.503208349644700, 4.207737267941194, 4.890365431657997, 5.262773659159644, 5.884356157093315, 6.668479466048576, 6.980378823231359, 7.555093606625250, 8.191347038913660, 8.491719587937563, 9.206746214918916, 9.632349046284574, 10.16325420990684, 10.78033232567414, 11.18407198350395, 11.66856904005622, 12.20118618857810, 12.65532903764547, 13.08657335456314, 13.72678963514116, 14.11678616051947
