| L(s) = 1 | + 2-s + 3-s + 4-s − 5-s + 6-s + 7-s + 8-s + 9-s − 10-s + 4·11-s + 12-s − 2·13-s + 14-s − 15-s + 16-s − 17-s + 18-s − 4·19-s − 20-s + 21-s + 4·22-s + 23-s + 24-s + 25-s − 2·26-s + 27-s + 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s + 1.20·11-s + 0.288·12-s − 0.554·13-s + 0.267·14-s − 0.258·15-s + 1/4·16-s − 0.242·17-s + 0.235·18-s − 0.917·19-s − 0.223·20-s + 0.218·21-s + 0.852·22-s + 0.208·23-s + 0.204·24-s + 1/5·25-s − 0.392·26-s + 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 82110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 82110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 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 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 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
−14.30473223918863, −13.72055310260169, −13.36631712559413, −12.69180248802346, −12.20794302019431, −11.93251897630439, −11.30836684300137, −10.85530087163134, −10.29580088098044, −9.678019625160252, −9.153883025341499, −8.436468794689817, −8.360546176934098, −7.352833474834433, −7.181569842367211, −6.506068693792925, −6.038966236719615, −5.262555798211373, −4.611167288465326, −4.230705263945579, −3.771918217522657, −3.032633056279264, −2.506643944441150, −1.736356392964314, −1.182186497088513, 0,
1.182186497088513, 1.736356392964314, 2.506643944441150, 3.032633056279264, 3.771918217522657, 4.230705263945579, 4.611167288465326, 5.262555798211373, 6.038966236719615, 6.506068693792925, 7.181569842367211, 7.352833474834433, 8.360546176934098, 8.436468794689817, 9.153883025341499, 9.678019625160252, 10.29580088098044, 10.85530087163134, 11.30836684300137, 11.93251897630439, 12.20794302019431, 12.69180248802346, 13.36631712559413, 13.72055310260169, 14.30473223918863
