L(s) = 1 | + (0.5 − 0.866i)2-s + (0.866 + 0.5i)3-s + (−0.499 − 0.866i)4-s + (0.866 − 0.499i)6-s + (0.758 + 1.31i)7-s − 0.999·8-s + (0.499 + 0.866i)9-s + (−4.45 − 2.57i)11-s − 0.999i·12-s + (1.86 − 3.08i)13-s + 1.51·14-s + (−0.5 + 0.866i)16-s + (4.15 − 2.39i)17-s + 0.999·18-s + (1.39 − 0.807i)19-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (0.499 + 0.288i)3-s + (−0.249 − 0.433i)4-s + (0.353 − 0.204i)6-s + (0.286 + 0.496i)7-s − 0.353·8-s + (0.166 + 0.288i)9-s + (−1.34 − 0.775i)11-s − 0.288i·12-s + (0.516 − 0.856i)13-s + 0.405·14-s + (−0.125 + 0.216i)16-s + (1.00 − 0.581i)17-s + 0.235·18-s + (0.320 − 0.185i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0663 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0663 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.235128674\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.235128674\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 + 0.866i)T \) |
| 3 | \( 1 + (-0.866 - 0.5i)T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + (-1.86 + 3.08i)T \) |
good | 7 | \( 1 + (-0.758 - 1.31i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (4.45 + 2.57i)T + (5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (-4.15 + 2.39i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.39 + 0.807i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (4.71 + 2.72i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-4.93 + 8.55i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 9.88iT - 31T^{2} \) |
| 37 | \( 1 + (-4.36 + 7.56i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.10 - 0.637i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-2.04 + 1.18i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 6.84T + 47T^{2} \) |
| 53 | \( 1 - 0.881iT - 53T^{2} \) |
| 59 | \( 1 + (-8.04 + 4.64i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (6.24 + 10.8i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (5.76 - 9.98i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-13.7 + 7.91i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 - 4.39T + 73T^{2} \) |
| 79 | \( 1 - 11.0T + 79T^{2} \) |
| 83 | \( 1 - 0.979T + 83T^{2} \) |
| 89 | \( 1 + (-3.54 - 2.04i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (2.85 + 4.95i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.032029392774872863351938222866, −8.021468439961909109699427891957, −7.967749061079951231443406549615, −6.38442312178014762236847239203, −5.45215800629661229248770903321, −5.02684714293252611770159381413, −3.78036990056367113233537519578, −2.97145806558802533348409756119, −2.30746907426642587573542048683, −0.70296541671444803453310681754,
1.41414527747597979536837462619, 2.62580517181091582533775461405, 3.70985588834321272869696564841, 4.49108650904340128429343776134, 5.40843433285436462693261793872, 6.27979948350561913314258826021, 7.15701784638002892143430423539, 7.917042994036979155628034110659, 8.146605752469532416530592368715, 9.346306976108898743522638117953