L(s) = 1 | − 2·3-s + (−2 − i)5-s − 2i·7-s + 9-s − 4i·11-s − 4·13-s + (4 + 2i)15-s − 4i·19-s + 4i·21-s − 2i·23-s + (3 + 4i)25-s + 4·27-s + 2i·29-s + 8i·33-s + (−2 + 4i)35-s + ⋯ |
L(s) = 1 | − 1.15·3-s + (−0.894 − 0.447i)5-s − 0.755i·7-s + 0.333·9-s − 1.20i·11-s − 1.10·13-s + (1.03 + 0.516i)15-s − 0.917i·19-s + 0.872i·21-s − 0.417i·23-s + (0.600 + 0.800i)25-s + 0.769·27-s + 0.371i·29-s + 1.39i·33-s + (−0.338 + 0.676i)35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.316 - 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.316 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + (2 + i)T \) |
good | 3 | \( 1 + 2T + 3T^{2} \) |
| 7 | \( 1 + 2iT - 7T^{2} \) |
| 11 | \( 1 + 4iT - 11T^{2} \) |
| 13 | \( 1 + 4T + 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 + 4iT - 19T^{2} \) |
| 23 | \( 1 + 2iT - 23T^{2} \) |
| 29 | \( 1 - 2iT - 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 4T + 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 + 6T + 43T^{2} \) |
| 47 | \( 1 + 6iT - 47T^{2} \) |
| 53 | \( 1 + 4T + 53T^{2} \) |
| 59 | \( 1 - 12iT - 59T^{2} \) |
| 61 | \( 1 - 10iT - 61T^{2} \) |
| 67 | \( 1 - 14T + 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 - 8iT - 73T^{2} \) |
| 79 | \( 1 + 16T + 79T^{2} \) |
| 83 | \( 1 + 2T + 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 - 16iT - 97T^{2} \) |
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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
−8.971032665478504390906883453618, −8.319573106054894694132332396057, −7.26363101655113282887250897072, −6.73541544353418112783135417997, −5.56456286812649005561726193789, −4.92842645001022242911447102592, −4.06098160677765844395542859410, −2.93093876388553492570718032090, −0.908202465642398068326577733774, 0,
2.00012656262799737057545119329, 3.25082624410801516081693744191, 4.52208046942585104835414977717, 5.13855625914583398066779660788, 6.09667991167548815023277447170, 6.93065976790340617406915001604, 7.62844616768852401829201966372, 8.507957154726195046580233896002, 9.706590691386421602897945019328