| L(s) = 1 | + (−0.173 + 0.984i)2-s + (−0.592 + 1.62i)3-s + (−0.939 − 0.342i)4-s + (−0.673 + 0.565i)5-s + (−1.5 − 0.866i)6-s + (3.31 − 1.20i)7-s + (0.5 − 0.866i)8-s + (−2.29 − 1.92i)9-s + (−0.439 − 0.761i)10-s + (2.73 + 2.29i)11-s + (1.11 − 1.32i)12-s + (−0.641 − 3.63i)13-s + (0.613 + 3.47i)14-s + (−0.520 − 1.43i)15-s + (0.766 + 0.642i)16-s + (−3.12 − 5.41i)17-s + ⋯ |
| L(s) = 1 | + (−0.122 + 0.696i)2-s + (−0.342 + 0.939i)3-s + (−0.469 − 0.171i)4-s + (−0.301 + 0.252i)5-s + (−0.612 − 0.353i)6-s + (1.25 − 0.456i)7-s + (0.176 − 0.306i)8-s + (−0.766 − 0.642i)9-s + (−0.139 − 0.240i)10-s + (0.825 + 0.692i)11-s + (0.321 − 0.383i)12-s + (−0.177 − 1.00i)13-s + (0.163 + 0.929i)14-s + (−0.134 − 0.369i)15-s + (0.191 + 0.160i)16-s + (−0.757 − 1.31i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0581 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0581 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.534690 + 0.504454i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.534690 + 0.504454i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.173 - 0.984i)T \) |
| 3 | \( 1 + (0.592 - 1.62i)T \) |
| good | 5 | \( 1 + (0.673 - 0.565i)T + (0.868 - 4.92i)T^{2} \) |
| 7 | \( 1 + (-3.31 + 1.20i)T + (5.36 - 4.49i)T^{2} \) |
| 11 | \( 1 + (-2.73 - 2.29i)T + (1.91 + 10.8i)T^{2} \) |
| 13 | \( 1 + (0.641 + 3.63i)T + (-12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (3.12 + 5.41i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (2.08 - 3.61i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.93 + 0.705i)T + (17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (-0.0282 + 0.160i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (1.53 + 0.560i)T + (23.7 + 19.9i)T^{2} \) |
| 37 | \( 1 + (-3.85 - 6.68i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (1.33 + 7.58i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (8.29 + 6.95i)T + (7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (-6.02 + 2.19i)T + (36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 - 0.716T + 53T^{2} \) |
| 59 | \( 1 + (5.35 - 4.49i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (-1.19 + 0.433i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (-0.624 - 3.54i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (-6.76 - 11.7i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-1.16 + 2.01i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (1.14 - 6.51i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (0.773 - 4.38i)T + (-77.9 - 28.3i)T^{2} \) |
| 89 | \( 1 + (4.62 - 8.00i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (8.64 + 7.25i)T + (16.8 + 95.5i)T^{2} \) |
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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
−15.44525257634354937000249238058, −14.88381728902378626140332815743, −13.89526051461352899044029549778, −11.99185661423792833650645331625, −10.92541084352993015091190066531, −9.782513291835111844702113889677, −8.406855106686713702850468837463, −7.08234984376031732557831993934, −5.31043601170801833382712253419, −4.13495627452181637836522602533,
1.86065738423948769723059802847, 4.51134062858418943789196767720, 6.31489566462143453866589393772, 8.083325919406855249335331831647, 8.905197427637646271298323099057, 11.05401582284760249851472027487, 11.57943101774857697278610835520, 12.58992602598772091875007465145, 13.82135480805910909119088915970, 14.78515912161169768974137448661
