L(s) = 1 | + (1.42 + 1.72i)5-s + (−3.11 − 3.11i)7-s + 1.17·11-s + (−2.15 − 2.15i)13-s + (−1.33 + 1.33i)17-s + 0.322·19-s + (−4.71 − 4.71i)23-s + (−0.933 + 4.91i)25-s − 6.63i·29-s + 0.0675·31-s + (0.924 − 9.82i)35-s + (−7.60 + 7.60i)37-s − 3.19i·41-s + (−6.70 − 6.70i)43-s + (−7.34 + 7.34i)47-s + ⋯ |
L(s) = 1 | + (0.637 + 0.770i)5-s + (−1.17 − 1.17i)7-s + 0.355·11-s + (−0.596 − 0.596i)13-s + (−0.323 + 0.323i)17-s + 0.0739·19-s + (−0.982 − 0.982i)23-s + (−0.186 + 0.982i)25-s − 1.23i·29-s + 0.0121·31-s + (0.156 − 1.66i)35-s + (−1.25 + 1.25i)37-s − 0.499i·41-s + (−1.02 − 1.02i)43-s + (−1.07 + 1.07i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.654 + 0.756i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.654 + 0.756i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6960404570\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6960404570\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-1.42 - 1.72i)T \) |
good | 7 | \( 1 + (3.11 + 3.11i)T + 7iT^{2} \) |
| 11 | \( 1 - 1.17T + 11T^{2} \) |
| 13 | \( 1 + (2.15 + 2.15i)T + 13iT^{2} \) |
| 17 | \( 1 + (1.33 - 1.33i)T - 17iT^{2} \) |
| 19 | \( 1 - 0.322T + 19T^{2} \) |
| 23 | \( 1 + (4.71 + 4.71i)T + 23iT^{2} \) |
| 29 | \( 1 + 6.63iT - 29T^{2} \) |
| 31 | \( 1 - 0.0675T + 31T^{2} \) |
| 37 | \( 1 + (7.60 - 7.60i)T - 37iT^{2} \) |
| 41 | \( 1 + 3.19iT - 41T^{2} \) |
| 43 | \( 1 + (6.70 + 6.70i)T + 43iT^{2} \) |
| 47 | \( 1 + (7.34 - 7.34i)T - 47iT^{2} \) |
| 53 | \( 1 + (-5.73 + 5.73i)T - 53iT^{2} \) |
| 59 | \( 1 + 8.68iT - 59T^{2} \) |
| 61 | \( 1 + 12.5iT - 61T^{2} \) |
| 67 | \( 1 + (-1.87 + 1.87i)T - 67iT^{2} \) |
| 71 | \( 1 + 4.18iT - 71T^{2} \) |
| 73 | \( 1 + (-3.97 + 3.97i)T - 73iT^{2} \) |
| 79 | \( 1 - 9.66iT - 79T^{2} \) |
| 83 | \( 1 + (-0.585 + 0.585i)T - 83iT^{2} \) |
| 89 | \( 1 - 0.557T + 89T^{2} \) |
| 97 | \( 1 + (10.5 + 10.5i)T + 97iT^{2} \) |
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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
−9.617212499204831440619165749852, −8.381789833257228770063920311529, −7.47890143865354635493029805189, −6.52964389136650246441053399454, −6.39084381687567411472199100694, −5.07059887878653598829847854999, −3.88061479586521760484642899502, −3.17802524719059219581691912115, −2.02252574987127537397893406334, −0.25801227952197030584607103718,
1.68045771771578602693284693707, 2.66506662263285207017677279864, 3.79699779210183655062864231328, 5.01191071487666836132077015027, 5.68406439773941078414950709937, 6.44669223092295660072177149117, 7.26690974195749043501271596579, 8.596685711155727494019179487990, 9.049529510915832041097472382604, 9.665622924901643461941010318881