| L(s) = 1 | − 3-s − 2·5-s + 9-s + 4·11-s − 2·13-s + 2·15-s + 6·17-s − 19-s + 4·23-s − 25-s − 27-s − 2·29-s + 4·31-s − 4·33-s + 10·37-s + 2·39-s − 10·41-s − 4·43-s − 2·45-s − 4·47-s − 6·51-s − 10·53-s − 8·55-s + 57-s + 12·59-s − 14·61-s + 4·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s + 1/3·9-s + 1.20·11-s − 0.554·13-s + 0.516·15-s + 1.45·17-s − 0.229·19-s + 0.834·23-s − 1/5·25-s − 0.192·27-s − 0.371·29-s + 0.718·31-s − 0.696·33-s + 1.64·37-s + 0.320·39-s − 1.56·41-s − 0.609·43-s − 0.298·45-s − 0.583·47-s − 0.840·51-s − 1.37·53-s − 1.07·55-s + 0.132·57-s + 1.56·59-s − 1.79·61-s + 0.496·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 44688 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 44688 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 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
−14.90210344907815, −14.64784545082200, −13.86563107280929, −13.35925019210962, −12.64790127277146, −12.12523090725393, −11.93609357210482, −11.29508820532116, −11.01152150130563, −10.12908651951271, −9.710307686450102, −9.298978648722878, −8.412192023048783, −8.001651613773108, −7.451825505355644, −6.855244019323992, −6.358561860778105, −5.767400287395972, −4.925480624639800, −4.658615806609393, −3.702540126012030, −3.508405002068279, −2.574371506777981, −1.529511277572048, −0.9416243638458639, 0,
0.9416243638458639, 1.529511277572048, 2.574371506777981, 3.508405002068279, 3.702540126012030, 4.658615806609393, 4.925480624639800, 5.767400287395972, 6.358561860778105, 6.855244019323992, 7.451825505355644, 8.001651613773108, 8.412192023048783, 9.298978648722878, 9.710307686450102, 10.12908651951271, 11.01152150130563, 11.29508820532116, 11.93609357210482, 12.12523090725393, 12.64790127277146, 13.35925019210962, 13.86563107280929, 14.64784545082200, 14.90210344907815
