| L(s) = 1 | + 3-s − 5-s + 9-s − 4·11-s − 13-s − 15-s + 17-s + 4·19-s + 25-s + 27-s + 2·29-s + 8·31-s − 4·33-s + 10·37-s − 39-s + 10·41-s − 4·43-s − 45-s − 8·47-s − 7·49-s + 51-s + 2·53-s + 4·55-s + 4·57-s + 4·59-s − 6·61-s + 65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1/3·9-s − 1.20·11-s − 0.277·13-s − 0.258·15-s + 0.242·17-s + 0.917·19-s + 1/5·25-s + 0.192·27-s + 0.371·29-s + 1.43·31-s − 0.696·33-s + 1.64·37-s − 0.160·39-s + 1.56·41-s − 0.609·43-s − 0.149·45-s − 1.16·47-s − 49-s + 0.140·51-s + 0.274·53-s + 0.539·55-s + 0.529·57-s + 0.520·59-s − 0.768·61-s + 0.124·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 212160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 212160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 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
−13.23325794429538, −12.86781257100564, −12.38218928980120, −11.73623977338995, −11.51596379935210, −10.88536520172847, −10.31379432751208, −10.00052225179360, −9.452862662982495, −9.104297224377940, −8.214851764051555, −8.073770155656788, −7.714291651666948, −7.176987442835427, −6.573448396905569, −6.039687249293252, −5.422501945239511, −4.846396738976629, −4.502314415686465, −3.859350835340382, −3.151548793051187, −2.765365488156739, −2.395507889852524, −1.424064674244481, −0.8518534840389159, 0,
0.8518534840389159, 1.424064674244481, 2.395507889852524, 2.765365488156739, 3.151548793051187, 3.859350835340382, 4.502314415686465, 4.846396738976629, 5.422501945239511, 6.039687249293252, 6.573448396905569, 7.176987442835427, 7.714291651666948, 8.073770155656788, 8.214851764051555, 9.104297224377940, 9.452862662982495, 10.00052225179360, 10.31379432751208, 10.88536520172847, 11.51596379935210, 11.73623977338995, 12.38218928980120, 12.86781257100564, 13.23325794429538
