| L(s) = 1 | − 2-s + 4-s + 5-s − 8-s − 10-s − 4·13-s + 16-s − 4·19-s + 20-s − 8·23-s + 25-s + 4·26-s + 2·29-s − 4·31-s − 32-s + 2·37-s + 4·38-s − 40-s − 10·41-s + 8·46-s + 12·47-s − 7·49-s − 50-s − 4·52-s − 2·53-s − 2·58-s − 8·59-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.353·8-s − 0.316·10-s − 1.10·13-s + 1/4·16-s − 0.917·19-s + 0.223·20-s − 1.66·23-s + 1/5·25-s + 0.784·26-s + 0.371·29-s − 0.718·31-s − 0.176·32-s + 0.328·37-s + 0.648·38-s − 0.158·40-s − 1.56·41-s + 1.17·46-s + 1.75·47-s − 49-s − 0.141·50-s − 0.554·52-s − 0.274·53-s − 0.262·58-s − 1.04·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 166410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 166410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3542494805\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3542494805\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 43 | \( 1 \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 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 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 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
−13.29857192381698, −12.55683047076170, −12.17106457365195, −12.02645695609376, −11.17075366861764, −10.76827341956766, −10.31878926292730, −9.899991810201552, −9.386954286045186, −9.112400860603512, −8.291662335218453, −8.076488561513054, −7.501930426380006, −6.952878413630780, −6.392487343376906, −6.083352353751422, −5.347679436445789, −4.909406628233818, −4.220580931052048, −3.664429352358325, −2.915522885097153, −2.254786365607078, −1.961503660761638, −1.218244887724657, −0.1912605729250838,
0.1912605729250838, 1.218244887724657, 1.961503660761638, 2.254786365607078, 2.915522885097153, 3.664429352358325, 4.220580931052048, 4.909406628233818, 5.347679436445789, 6.083352353751422, 6.392487343376906, 6.952878413630780, 7.501930426380006, 8.076488561513054, 8.291662335218453, 9.112400860603512, 9.386954286045186, 9.899991810201552, 10.31878926292730, 10.76827341956766, 11.17075366861764, 12.02645695609376, 12.17106457365195, 12.55683047076170, 13.29857192381698
