| L(s) = 1 | + (1.23 + 0.866i)2-s + (1.45 + 0.938i)3-s + (0.0960 + 0.264i)4-s + (−2.23 − 0.140i)5-s + (0.987 + 2.42i)6-s + (0.677 + 1.45i)7-s + (0.671 − 2.50i)8-s + (1.23 + 2.73i)9-s + (−2.63 − 2.10i)10-s + (−1.78 − 2.12i)11-s + (−0.107 + 0.474i)12-s + (0.176 + 0.251i)13-s + (−0.420 + 2.38i)14-s + (−3.11 − 2.29i)15-s + (3.43 − 2.88i)16-s + (−1.88 − 7.02i)17-s + ⋯ |
| L(s) = 1 | + (0.874 + 0.612i)2-s + (0.840 + 0.542i)3-s + (0.0480 + 0.132i)4-s + (−0.998 − 0.0626i)5-s + (0.403 + 0.988i)6-s + (0.256 + 0.549i)7-s + (0.237 − 0.886i)8-s + (0.412 + 0.911i)9-s + (−0.834 − 0.666i)10-s + (−0.538 − 0.641i)11-s + (−0.0311 + 0.136i)12-s + (0.0489 + 0.0698i)13-s + (−0.112 + 0.637i)14-s + (−0.804 − 0.593i)15-s + (0.858 − 0.720i)16-s + (−0.456 − 1.70i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.601 - 0.798i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.601 - 0.798i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.57835 + 0.786817i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.57835 + 0.786817i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.45 - 0.938i)T \) |
| 5 | \( 1 + (2.23 + 0.140i)T \) |
| good | 2 | \( 1 + (-1.23 - 0.866i)T + (0.684 + 1.87i)T^{2} \) |
| 7 | \( 1 + (-0.677 - 1.45i)T + (-4.49 + 5.36i)T^{2} \) |
| 11 | \( 1 + (1.78 + 2.12i)T + (-1.91 + 10.8i)T^{2} \) |
| 13 | \( 1 + (-0.176 - 0.251i)T + (-4.44 + 12.2i)T^{2} \) |
| 17 | \( 1 + (1.88 + 7.02i)T + (-14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (5.07 - 2.92i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.116 - 0.0543i)T + (14.7 + 17.6i)T^{2} \) |
| 29 | \( 1 + (-1.51 - 8.60i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (-4.22 + 1.53i)T + (23.7 - 19.9i)T^{2} \) |
| 37 | \( 1 + (-2.08 + 0.558i)T + (32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + (4.62 + 0.815i)T + (38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (0.429 - 4.90i)T + (-42.3 - 7.46i)T^{2} \) |
| 47 | \( 1 + (7.69 - 3.58i)T + (30.2 - 36.0i)T^{2} \) |
| 53 | \( 1 + (-0.483 - 0.483i)T + 53iT^{2} \) |
| 59 | \( 1 + (-7.43 - 6.23i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (-2.50 - 0.910i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (0.884 - 0.619i)T + (22.9 - 62.9i)T^{2} \) |
| 71 | \( 1 + (-9.03 - 5.21i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (6.65 + 1.78i)T + (63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (-2.04 + 0.360i)T + (74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (-7.55 + 10.7i)T + (-28.3 - 77.9i)T^{2} \) |
| 89 | \( 1 + (5.58 + 9.66i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (11.8 + 1.03i)T + (95.5 + 16.8i)T^{2} \) |
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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
−13.62537878397687345065301909890, −12.71188633126798812952522525831, −11.44360661361224125501381872770, −10.29515467557863629122933884777, −8.963848025380710327798082982223, −8.060755697707632286619460716062, −6.88125268920970164403866077481, −5.24444286405205228966191667310, −4.36997140550704350808566354826, −3.05596020943065217722975410273,
2.28129387235537543309900685011, 3.76132183687117355768146141369, 4.52403998257154271711879632009, 6.66031329119992133247482563521, 7.980966924977533897847206588889, 8.434254948779446178221205593201, 10.30811441500191914653909252077, 11.32697842579899055954375635528, 12.39647399702147550634530959915, 12.99463898269683553431349615397
