| L(s) = 1 | + (−0.786 − 0.618i)2-s + (1.77 − 2.48i)3-s + (0.235 + 0.971i)4-s + (0.592 − 0.114i)5-s + (−2.93 + 0.861i)6-s + (1.79 − 1.94i)7-s + (0.415 − 0.909i)8-s + (−2.07 − 5.99i)9-s + (−0.536 − 0.276i)10-s + (−4.66 + 3.67i)11-s + (2.83 + 1.13i)12-s + (2.06 − 1.32i)13-s + (−2.61 + 0.423i)14-s + (0.766 − 1.67i)15-s + (−0.888 + 0.458i)16-s + (3.26 + 3.11i)17-s + ⋯ |
| L(s) = 1 | + (−0.555 − 0.437i)2-s + (1.02 − 1.43i)3-s + (0.117 + 0.485i)4-s + (0.265 − 0.0510i)5-s + (−1.19 + 0.351i)6-s + (0.676 − 0.736i)7-s + (0.146 − 0.321i)8-s + (−0.691 − 1.99i)9-s + (−0.169 − 0.0874i)10-s + (−1.40 + 1.10i)11-s + (0.819 + 0.327i)12-s + (0.572 − 0.367i)13-s + (−0.697 + 0.113i)14-s + (0.197 − 0.433i)15-s + (−0.222 + 0.114i)16-s + (0.792 + 0.755i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.454 + 0.890i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.454 + 0.890i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.758462 - 1.23867i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.758462 - 1.23867i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.786 + 0.618i)T \) |
| 7 | \( 1 + (-1.79 + 1.94i)T \) |
| 23 | \( 1 + (-4.73 - 0.772i)T \) |
| good | 3 | \( 1 + (-1.77 + 2.48i)T + (-0.981 - 2.83i)T^{2} \) |
| 5 | \( 1 + (-0.592 + 0.114i)T + (4.64 - 1.85i)T^{2} \) |
| 11 | \( 1 + (4.66 - 3.67i)T + (2.59 - 10.6i)T^{2} \) |
| 13 | \( 1 + (-2.06 + 1.32i)T + (5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (-3.26 - 3.11i)T + (0.808 + 16.9i)T^{2} \) |
| 19 | \( 1 + (-2.33 + 2.22i)T + (0.904 - 18.9i)T^{2} \) |
| 29 | \( 1 + (4.81 - 1.41i)T + (24.3 - 15.6i)T^{2} \) |
| 31 | \( 1 + (7.43 - 0.709i)T + (30.4 - 5.86i)T^{2} \) |
| 37 | \( 1 + (-0.372 - 1.07i)T + (-29.0 + 22.8i)T^{2} \) |
| 41 | \( 1 + (-5.23 - 6.04i)T + (-5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (2.79 + 6.12i)T + (-28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 + (-2.59 + 4.49i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (0.457 - 9.60i)T + (-52.7 - 5.03i)T^{2} \) |
| 59 | \( 1 + (-7.43 - 3.83i)T + (34.2 + 48.0i)T^{2} \) |
| 61 | \( 1 + (-2.81 - 3.94i)T + (-19.9 + 57.6i)T^{2} \) |
| 67 | \( 1 + (-11.4 + 4.57i)T + (48.4 - 46.2i)T^{2} \) |
| 71 | \( 1 + (0.0672 - 0.468i)T + (-68.1 - 20.0i)T^{2} \) |
| 73 | \( 1 + (-0.143 - 0.592i)T + (-64.8 + 33.4i)T^{2} \) |
| 79 | \( 1 + (-0.415 - 8.72i)T + (-78.6 + 7.50i)T^{2} \) |
| 83 | \( 1 + (-4.87 + 5.62i)T + (-11.8 - 82.1i)T^{2} \) |
| 89 | \( 1 + (0.281 + 0.0268i)T + (87.3 + 16.8i)T^{2} \) |
| 97 | \( 1 + (-0.356 - 0.411i)T + (-13.8 + 96.0i)T^{2} \) |
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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
−11.30306516489527651231236539195, −10.35451243041858281829551635285, −9.322734269293386239791467960313, −8.273674194860923328040433465435, −7.56632810884632356777783894198, −7.14330635550855643860135940881, −5.43954986832651317367142205069, −3.59114509231746785411542078418, −2.30460313929121008808040503025, −1.28231246160990091615267466945,
2.39145634890736965028905397732, 3.56605276396570399120874008496, 5.16092012564773642866521904007, 5.65499195203974118758865137054, 7.69581740502999399558943299080, 8.294135821133989035391643919251, 9.142885628583937259679894770123, 9.777289344810354614636597603601, 10.81623220072586059725226753105, 11.36212393264837090148216952662
