| L(s) = 1 | − 3-s + 5-s + 7-s + 9-s − 4·11-s − 2·13-s − 15-s + 2·17-s + 4·19-s − 21-s + 23-s + 25-s − 27-s − 2·29-s − 8·31-s + 4·33-s + 35-s + 6·37-s + 2·39-s + 2·41-s − 4·43-s + 45-s − 8·47-s + 49-s − 2·51-s − 2·53-s − 4·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.377·7-s + 1/3·9-s − 1.20·11-s − 0.554·13-s − 0.258·15-s + 0.485·17-s + 0.917·19-s − 0.218·21-s + 0.208·23-s + 1/5·25-s − 0.192·27-s − 0.371·29-s − 1.43·31-s + 0.696·33-s + 0.169·35-s + 0.986·37-s + 0.320·39-s + 0.312·41-s − 0.609·43-s + 0.149·45-s − 1.16·47-s + 1/7·49-s − 0.280·51-s − 0.274·53-s − 0.539·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 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 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−14.98647612613533, −14.55634078931633, −14.16757913387242, −13.33113010141543, −12.96535868184717, −12.65737408184639, −11.81564475204883, −11.44320429493830, −10.90788994191993, −10.37380302931553, −9.710065923388775, −9.578357389939764, −8.634341330632794, −8.037476116482024, −7.483738685643219, −7.088987650277144, −6.312908327159483, −5.560409587589385, −5.288045424364336, −4.837819549358456, −3.959782892890469, −3.203076987879458, −2.490437913529518, −1.806363170631562, −0.9528052808987018, 0,
0.9528052808987018, 1.806363170631562, 2.490437913529518, 3.203076987879458, 3.959782892890469, 4.837819549358456, 5.288045424364336, 5.560409587589385, 6.312908327159483, 7.088987650277144, 7.483738685643219, 8.037476116482024, 8.634341330632794, 9.578357389939764, 9.710065923388775, 10.37380302931553, 10.90788994191993, 11.44320429493830, 11.81564475204883, 12.65737408184639, 12.96535868184717, 13.33113010141543, 14.16757913387242, 14.55634078931633, 14.98647612613533
