L(s) = 1 | + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s + (−1.26 + 1.84i)5-s + (−2.17 + 3.76i)7-s − 0.999·8-s + (−2.22 − 0.178i)10-s + (2.04 − 1.17i)11-s + (−3.18 + 1.69i)13-s − 4.34·14-s + (−0.5 − 0.866i)16-s + (−2.60 − 1.50i)17-s + (0.585 + 0.338i)19-s + (−0.960 − 2.01i)20-s + (2.04 + 1.17i)22-s + (5.58 − 3.22i)23-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.567 + 0.823i)5-s + (−0.821 + 1.42i)7-s − 0.353·8-s + (−0.704 − 0.0563i)10-s + (0.615 − 0.355i)11-s + (−0.883 + 0.469i)13-s − 1.16·14-s + (−0.125 − 0.216i)16-s + (−0.631 − 0.364i)17-s + (0.134 + 0.0776i)19-s + (−0.214 − 0.451i)20-s + (0.435 + 0.251i)22-s + (1.16 − 0.672i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.606 + 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.606 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6277833128\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6277833128\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 + (1.26 - 1.84i)T \) |
| 13 | \( 1 + (3.18 - 1.69i)T \) |
good | 7 | \( 1 + (2.17 - 3.76i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.04 + 1.17i)T + (5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (2.60 + 1.50i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.585 - 0.338i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-5.58 + 3.22i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (4.82 + 8.35i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 7.11iT - 31T^{2} \) |
| 37 | \( 1 + (-3.74 - 6.48i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-2.60 + 1.50i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (5.91 + 3.41i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 5.61T + 47T^{2} \) |
| 53 | \( 1 - 9.43iT - 53T^{2} \) |
| 59 | \( 1 + (-4.56 - 2.63i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.15 + 3.73i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (2.91 + 5.04i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (2.52 + 1.45i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + 7.67T + 73T^{2} \) |
| 79 | \( 1 + 3.74T + 79T^{2} \) |
| 83 | \( 1 + 10.3T + 83T^{2} \) |
| 89 | \( 1 + (4.15 - 2.39i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (8.17 - 14.1i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.15853037485789551194370294626, −9.258638488113208432506904929615, −8.734502175509939769172901301262, −7.69779790494387532478234148829, −6.71817727573123320083764717655, −6.39181669639872892829185136254, −5.32271695704096431057486192550, −4.30796620732110268767826390851, −3.16048142878067037385001348142, −2.47920367500569096094810180775,
0.24210824479611131174410567684, 1.47223673817556709181586449324, 3.14147148773562638709949672389, 3.97352182822856393623716026296, 4.62891092916121156391852164909, 5.63421456858660865260255650621, 6.97613456361520624935907905345, 7.38427127130390003005095165787, 8.590127714364853257072661147881, 9.545066743215879253582836051239