| L(s) = 1 | + (−1.37 + 0.323i)2-s + (0.987 − 0.156i)3-s + (1.79 − 0.889i)4-s + (−2.11 − 0.728i)5-s + (−1.30 + 0.534i)6-s + (0.284 + 0.284i)7-s + (−2.17 + 1.80i)8-s + (0.951 − 0.309i)9-s + (3.14 + 0.319i)10-s + (4.58 + 1.49i)11-s + (1.63 − 1.15i)12-s + (2.80 − 5.51i)13-s + (−0.482 − 0.299i)14-s + (−2.20 − 0.388i)15-s + (2.41 − 3.18i)16-s + (−0.127 + 0.807i)17-s + ⋯ |
| L(s) = 1 | + (−0.973 + 0.228i)2-s + (0.570 − 0.0903i)3-s + (0.895 − 0.444i)4-s + (−0.945 − 0.325i)5-s + (−0.534 + 0.218i)6-s + (0.107 + 0.107i)7-s + (−0.770 + 0.637i)8-s + (0.317 − 0.103i)9-s + (0.994 + 0.101i)10-s + (1.38 + 0.449i)11-s + (0.470 − 0.334i)12-s + (0.778 − 1.52i)13-s + (−0.129 − 0.0800i)14-s + (−0.568 − 0.100i)15-s + (0.604 − 0.796i)16-s + (−0.0310 + 0.195i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.912 + 0.410i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.912 + 0.410i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.945527 - 0.202814i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.945527 - 0.202814i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.37 - 0.323i)T \) |
| 3 | \( 1 + (-0.987 + 0.156i)T \) |
| 5 | \( 1 + (2.11 + 0.728i)T \) |
| good | 7 | \( 1 + (-0.284 - 0.284i)T + 7iT^{2} \) |
| 11 | \( 1 + (-4.58 - 1.49i)T + (8.89 + 6.46i)T^{2} \) |
| 13 | \( 1 + (-2.80 + 5.51i)T + (-7.64 - 10.5i)T^{2} \) |
| 17 | \( 1 + (0.127 - 0.807i)T + (-16.1 - 5.25i)T^{2} \) |
| 19 | \( 1 + (-1.06 + 0.776i)T + (5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + (2.81 + 5.52i)T + (-13.5 + 18.6i)T^{2} \) |
| 29 | \( 1 + (-4.62 + 6.36i)T + (-8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (-1.92 - 2.65i)T + (-9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (-4.49 - 2.29i)T + (21.7 + 29.9i)T^{2} \) |
| 41 | \( 1 + (-0.177 - 0.545i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + (8.22 - 8.22i)T - 43iT^{2} \) |
| 47 | \( 1 + (0.342 + 2.16i)T + (-44.6 + 14.5i)T^{2} \) |
| 53 | \( 1 + (-0.486 - 3.07i)T + (-50.4 + 16.3i)T^{2} \) |
| 59 | \( 1 + (-2.94 - 9.05i)T + (-47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (3.92 - 12.0i)T + (-49.3 - 35.8i)T^{2} \) |
| 67 | \( 1 + (10.4 + 1.66i)T + (63.7 + 20.7i)T^{2} \) |
| 71 | \( 1 + (-0.421 + 0.580i)T + (-21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (0.125 - 0.0641i)T + (42.9 - 59.0i)T^{2} \) |
| 79 | \( 1 + (0.419 + 0.305i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (0.432 - 2.73i)T + (-78.9 - 25.6i)T^{2} \) |
| 89 | \( 1 + (11.2 + 3.66i)T + (72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (9.56 - 1.51i)T + (92.2 - 29.9i)T^{2} \) |
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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
−11.68618711180274958827428006806, −10.56257996336500658002871510088, −9.667716998705180381450508263943, −8.459533398801178729596199626850, −8.252429130690853218628569672691, −7.08639234509575874538130306215, −6.06479378681265740675226171690, −4.36646701942332144516476408592, −2.98575446341531771134726048249, −1.10426160209441587579857788587,
1.53427843542514417161923275033, 3.33919746444719031184692363840, 4.11613031598849790566513293021, 6.40270702294338755319468753247, 7.13474081230678338124402180934, 8.204691262402674695409142923608, 8.941379448811379722945261331213, 9.699605121096582892676831395005, 11.01471100259969167900897209796, 11.56202296349523216366161441553
