| L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s − 7-s − 8-s + 9-s − 10-s + 3·11-s − 12-s + 2·13-s + 14-s − 15-s + 16-s − 17-s − 18-s − 19-s + 20-s + 21-s − 3·22-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 + 0.904·11-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 − 0.229·19-s + 0.223·20-s + 0.218·21-s − 0.639·22-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 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 - 9 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 + 9 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 - 11 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 + 11 T + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
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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
−14.12541673227984, −13.55140515087466, −13.40790657420857, −12.51275149014399, −12.13462247880189, −11.76260861766436, −11.09387419566167, −10.71805320410787, −10.23505736242731, −9.604743967038337, −9.367395497522360, −8.664796981538845, −8.310970267610410, −7.558799504237692, −6.999005054144196, −6.371385483175066, −6.224391332487146, −5.650076650914217, −4.792135563481625, −4.290814165137145, −3.638160182130162, −2.807128574356339, −2.323396680980555, −1.263092767697954, −1.057469327412862, 0,
1.057469327412862, 1.263092767697954, 2.323396680980555, 2.807128574356339, 3.638160182130162, 4.290814165137145, 4.792135563481625, 5.650076650914217, 6.224391332487146, 6.371385483175066, 6.999005054144196, 7.558799504237692, 8.310970267610410, 8.664796981538845, 9.367395497522360, 9.604743967038337, 10.23505736242731, 10.71805320410787, 11.09387419566167, 11.76260861766436, 12.13462247880189, 12.51275149014399, 13.40790657420857, 13.55140515087466, 14.12541673227984
