| L(s) = 1 | + (1.37 + 0.292i)2-s + (1.63 − 0.572i)3-s + (−0.0128 − 0.00572i)4-s + (1.56 − 1.59i)5-s + (2.42 − 0.310i)6-s + (−2.48 + 4.30i)7-s + (−2.29 − 1.66i)8-s + (2.34 − 1.87i)9-s + (2.62 − 1.74i)10-s + (3.07 + 0.653i)11-s + (−0.0242 − 0.00199i)12-s + (−1.44 + 0.307i)13-s + (−4.68 + 5.20i)14-s + (1.63 − 3.50i)15-s + (−2.65 − 2.95i)16-s + (−2.39 − 1.73i)17-s + ⋯ |
| L(s) = 1 | + (0.974 + 0.207i)2-s + (0.943 − 0.330i)3-s + (−0.00642 − 0.00286i)4-s + (0.698 − 0.715i)5-s + (0.988 − 0.126i)6-s + (−0.939 + 1.62i)7-s + (−0.811 − 0.589i)8-s + (0.781 − 0.624i)9-s + (0.829 − 0.552i)10-s + (0.926 + 0.196i)11-s + (−0.00701 − 0.000574i)12-s + (−0.401 + 0.0852i)13-s + (−1.25 + 1.39i)14-s + (0.422 − 0.906i)15-s + (−0.664 − 0.737i)16-s + (−0.579 − 0.421i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 + 0.146i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.989 + 0.146i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.31760 - 0.170640i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.31760 - 0.170640i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.63 + 0.572i)T \) |
| 5 | \( 1 + (-1.56 + 1.59i)T \) |
| good | 2 | \( 1 + (-1.37 - 0.292i)T + (1.82 + 0.813i)T^{2} \) |
| 7 | \( 1 + (2.48 - 4.30i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-3.07 - 0.653i)T + (10.0 + 4.47i)T^{2} \) |
| 13 | \( 1 + (1.44 - 0.307i)T + (11.8 - 5.28i)T^{2} \) |
| 17 | \( 1 + (2.39 + 1.73i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (1.51 + 1.09i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (4.45 - 4.94i)T + (-2.40 - 22.8i)T^{2} \) |
| 29 | \( 1 + (-0.427 - 4.06i)T + (-28.3 + 6.02i)T^{2} \) |
| 31 | \( 1 + (0.681 - 6.47i)T + (-30.3 - 6.44i)T^{2} \) |
| 37 | \( 1 + (-1.50 + 4.62i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (-2.38 + 0.508i)T + (37.4 - 16.6i)T^{2} \) |
| 43 | \( 1 + (1.92 - 3.32i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (0.175 + 1.66i)T + (-45.9 + 9.77i)T^{2} \) |
| 53 | \( 1 + (-0.228 + 0.165i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (-2.95 + 0.628i)T + (53.8 - 23.9i)T^{2} \) |
| 61 | \( 1 + (-7.58 - 1.61i)T + (55.7 + 24.8i)T^{2} \) |
| 67 | \( 1 + (-1.19 + 11.3i)T + (-65.5 - 13.9i)T^{2} \) |
| 71 | \( 1 + (-13.0 + 9.48i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (2.03 + 6.27i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (0.847 + 8.05i)T + (-77.2 + 16.4i)T^{2} \) |
| 83 | \( 1 + (8.79 - 3.91i)T + (55.5 - 61.6i)T^{2} \) |
| 89 | \( 1 + (-0.845 - 2.60i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (-1.02 - 9.77i)T + (-94.8 + 20.1i)T^{2} \) |
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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
−12.47722561456311461542366442072, −12.01374007327691190620175103561, −9.677764422542375919337614092166, −9.279139818695227121977177456975, −8.628069918708017237615859416400, −6.77770299914015207163013174293, −5.98595082073876292891746879050, −4.88661984508902848755460951733, −3.48415852635062427733166558886, −2.17961953511117110306359509815,
2.50387077765487573436184945049, 3.76457528462634113470042137279, 4.26245835247463382675933003873, 6.13780753411361562465083194430, 7.00524019539395298404715897090, 8.368914382927969009843289822972, 9.659097339988412640339219275100, 10.15218572836180032051909179437, 11.28439934632494687537128157693, 12.82790580454708629356058458437
