L(s) = 1 | + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s − 5-s + 0.999·8-s + (−0.5 − 0.866i)9-s + (0.5 − 0.866i)10-s + (−0.5 + 0.866i)13-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)17-s + 0.999·18-s + (0.499 + 0.866i)20-s + (−0.499 − 0.866i)26-s + (0.5 − 0.866i)29-s + (−0.499 − 0.866i)32-s − 0.999·34-s + ⋯ |
L(s) = 1 | + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s − 5-s + 0.999·8-s + (−0.5 − 0.866i)9-s + (0.5 − 0.866i)10-s + (−0.5 + 0.866i)13-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)17-s + 0.999·18-s + (0.499 + 0.866i)20-s + (−0.499 − 0.866i)26-s + (0.5 − 0.866i)29-s + (−0.499 − 0.866i)32-s − 0.999·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 52 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.711 - 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 52 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.711 - 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3248332447\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3248332447\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 13 | \( 1 + (0.5 - 0.866i)T \) |
good | 3 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 5 | \( 1 + T + T^{2} \) |
| 7 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + T + T^{2} \) |
| 59 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 73 | \( 1 + T + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
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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
−15.86901024324633578237109774276, −14.96124058887919042375486728325, −14.12767115902838684115562210190, −12.40865268975567652680377933239, −11.24632239324280198745131969122, −9.709118972498971638443287007627, −8.532649398844231199448686039577, −7.39766540537995423654738286953, −6.04668941269644592254494882943, −4.16168360291039984927090654039,
3.05212385545771777947032732532, 4.87537373101459305382914912093, 7.53136053749764317001096925301, 8.339948637340426939401113737946, 9.896784789924800205671186547144, 11.08546064930377023602876475562, 11.93591325352333454299037842375, 13.06765961975395203124255805669, 14.36669503053971576318634699798, 15.87294873793952215742661356483