L(s) = 1 | + (0.746 + 1.29i)2-s + (−1.73 − 0.0231i)3-s + (−0.114 + 0.197i)4-s + (0.5 − 0.866i)5-s + (−1.26 − 2.25i)6-s + (−0.5 − 0.866i)7-s + 2.64·8-s + (2.99 + 0.0800i)9-s + 1.49·10-s + (2.28 + 3.95i)11-s + (0.202 − 0.339i)12-s + (1.66 − 2.89i)13-s + (0.746 − 1.29i)14-s + (−0.885 + 1.48i)15-s + (2.20 + 3.81i)16-s − 0.168·17-s + ⋯ |
L(s) = 1 | + (0.527 + 0.914i)2-s + (−0.999 − 0.0133i)3-s + (−0.0570 + 0.0987i)4-s + (0.223 − 0.387i)5-s + (−0.515 − 0.921i)6-s + (−0.188 − 0.327i)7-s + 0.935·8-s + (0.999 + 0.0266i)9-s + 0.472·10-s + (0.688 + 1.19i)11-s + (0.0583 − 0.0979i)12-s + (0.462 − 0.801i)13-s + (0.199 − 0.345i)14-s + (−0.228 + 0.384i)15-s + (0.550 + 0.953i)16-s − 0.0408·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.782 - 0.622i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.782 - 0.622i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.41952 + 0.495316i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.41952 + 0.495316i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.73 + 0.0231i)T \) |
| 5 | \( 1 + (-0.5 + 0.866i)T \) |
| 7 | \( 1 + (0.5 + 0.866i)T \) |
good | 2 | \( 1 + (-0.746 - 1.29i)T + (-1 + 1.73i)T^{2} \) |
| 11 | \( 1 + (-2.28 - 3.95i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.66 + 2.89i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + 0.168T + 17T^{2} \) |
| 19 | \( 1 - 3.03T + 19T^{2} \) |
| 23 | \( 1 + (-1.72 + 2.98i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.731 - 1.26i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (3.89 - 6.74i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 9.18T + 37T^{2} \) |
| 41 | \( 1 + (-4.73 + 8.19i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (3.83 + 6.63i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-1.79 - 3.10i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 10.8T + 53T^{2} \) |
| 59 | \( 1 + (2.49 - 4.31i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (3.76 + 6.51i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (6.25 - 10.8i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 8.27T + 71T^{2} \) |
| 73 | \( 1 + 14.1T + 73T^{2} \) |
| 79 | \( 1 + (-1.32 - 2.28i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-2.80 - 4.86i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 9.53T + 89T^{2} \) |
| 97 | \( 1 + (1.59 + 2.76i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−12.00324678687383376204773422880, −10.64294901367829844611417794532, −10.20212277200872404414727292196, −8.885174758986394735321254702064, −7.31502392773909206222108685233, −6.89217120697862276019142514386, −5.74402897669941421125538178043, −5.04953649302334547847091295346, −4.01956401745477659940081619847, −1.40739149159242040001087936626,
1.50962758201706958086134527201, 3.19567666835508536729759667203, 4.21275234868909787525734408746, 5.54939681622524766976989446362, 6.44184540686408864484449972935, 7.50397657227417561538249915185, 9.035849191983495711836258235576, 10.06028993018635711903647174401, 11.10311293345221369546300194219, 11.49516941849396340688456124452