L(s) = 1 | + (0.781 + 0.623i)2-s + (−1.36 + 2.82i)3-s + (0.222 + 0.974i)4-s + (−0.747 + 2.10i)5-s + (−2.82 + 1.36i)6-s + (−1.52 − 2.16i)7-s + (−0.433 + 0.900i)8-s + (−4.27 − 5.35i)9-s + (−1.89 + 1.18i)10-s + (−0.773 + 0.969i)11-s + (−3.05 − 0.698i)12-s + (4.30 + 3.43i)13-s + (0.153 − 2.64i)14-s + (−4.94 − 4.98i)15-s + (−0.900 + 0.433i)16-s + (−1.89 − 0.431i)17-s + ⋯ |
L(s) = 1 | + (0.552 + 0.440i)2-s + (−0.786 + 1.63i)3-s + (0.111 + 0.487i)4-s + (−0.334 + 0.942i)5-s + (−1.15 + 0.555i)6-s + (−0.577 − 0.816i)7-s + (−0.153 + 0.318i)8-s + (−1.42 − 1.78i)9-s + (−0.600 + 0.373i)10-s + (−0.233 + 0.292i)11-s + (−0.883 − 0.201i)12-s + (1.19 + 0.951i)13-s + (0.0409 − 0.705i)14-s + (−1.27 − 1.28i)15-s + (−0.225 + 0.108i)16-s + (−0.458 − 0.104i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.723 + 0.690i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.723 + 0.690i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.340983 - 0.851851i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.340983 - 0.851851i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.781 - 0.623i)T \) |
| 5 | \( 1 + (0.747 - 2.10i)T \) |
| 7 | \( 1 + (1.52 + 2.16i)T \) |
good | 3 | \( 1 + (1.36 - 2.82i)T + (-1.87 - 2.34i)T^{2} \) |
| 11 | \( 1 + (0.773 - 0.969i)T + (-2.44 - 10.7i)T^{2} \) |
| 13 | \( 1 + (-4.30 - 3.43i)T + (2.89 + 12.6i)T^{2} \) |
| 17 | \( 1 + (1.89 + 0.431i)T + (15.3 + 7.37i)T^{2} \) |
| 19 | \( 1 - 2.48T + 19T^{2} \) |
| 23 | \( 1 + (3.40 - 0.778i)T + (20.7 - 9.97i)T^{2} \) |
| 29 | \( 1 + (0.423 - 1.85i)T + (-26.1 - 12.5i)T^{2} \) |
| 31 | \( 1 + 0.154T + 31T^{2} \) |
| 37 | \( 1 + (-5.27 - 1.20i)T + (33.3 + 16.0i)T^{2} \) |
| 41 | \( 1 + (-6.23 - 3.00i)T + (25.5 + 32.0i)T^{2} \) |
| 43 | \( 1 + (0.374 + 0.778i)T + (-26.8 + 33.6i)T^{2} \) |
| 47 | \( 1 + (7.75 + 6.18i)T + (10.4 + 45.8i)T^{2} \) |
| 53 | \( 1 + (10.8 - 2.47i)T + (47.7 - 22.9i)T^{2} \) |
| 59 | \( 1 + (13.2 - 6.37i)T + (36.7 - 46.1i)T^{2} \) |
| 61 | \( 1 + (2.42 - 10.6i)T + (-54.9 - 26.4i)T^{2} \) |
| 67 | \( 1 - 10.7iT - 67T^{2} \) |
| 71 | \( 1 + (2.64 + 11.6i)T + (-63.9 + 30.8i)T^{2} \) |
| 73 | \( 1 + (-5.96 + 4.75i)T + (16.2 - 71.1i)T^{2} \) |
| 79 | \( 1 - 12.9T + 79T^{2} \) |
| 83 | \( 1 + (7.66 - 6.11i)T + (18.4 - 80.9i)T^{2} \) |
| 89 | \( 1 + (-7.92 - 9.93i)T + (-19.8 + 86.7i)T^{2} \) |
| 97 | \( 1 + 1.61iT - 97T^{2} \) |
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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
−11.25060087577950950541132897786, −10.78116928277668077405355726344, −9.908513234992710275293311769613, −9.113031194749399384512627604427, −7.68775296024571278471035196072, −6.53157122807092516503652912427, −6.03021301043396083809799846018, −4.64464374339646727568858417392, −3.96203041317873502923151612420, −3.20185707242156532958846724733,
0.50631359380687349440454918454, 1.81528678160373617555803916692, 3.21978252326684058739586305295, 4.90910485420995406379833186202, 5.95814501885098611266362262755, 6.20169184537816053041070523728, 7.72720007245274279081554518022, 8.348900704754140948258388927196, 9.499403209259159486012837345720, 11.00082029090964869287970557512