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
−9.706590691386421602897945019328, −8.507957154726195046580233896002, −7.62844616768852401829201966372, −6.93065976790340617406915001604, −6.09667991167548815023277447170, −5.13855625914583398066779660788, −4.52208046942585104835414977717, −3.25082624410801516081693744191, −2.00012656262799737057545119329, 0,
0.908202465642398068326577733774, 2.93093876388553492570718032090, 4.06098160677765844395542859410, 4.92842645001022242911447102592, 5.56456286812649005561726193789, 6.73541544353418112783135417997, 7.26363101655113282887250897072, 8.319573106054894694132332396057, 8.971032665478504390906883453618