| L(s) = 1 | + (−0.109 + 1.40i)2-s + (−1.32 − 0.902i)3-s + (−1.97 − 0.309i)4-s + (0.323 + 0.0242i)5-s + (1.41 − 1.76i)6-s + (−1.28 − 2.31i)7-s + (0.653 − 2.75i)8-s + (−0.159 − 0.405i)9-s + (−0.0697 + 0.453i)10-s + (−2.61 − 1.02i)11-s + (2.33 + 2.19i)12-s + (2.01 − 1.60i)13-s + (3.40 − 1.55i)14-s + (−0.406 − 0.323i)15-s + (3.80 + 1.22i)16-s + (−0.295 − 0.317i)17-s + ⋯ |
| L(s) = 1 | + (−0.0776 + 0.996i)2-s + (−0.763 − 0.520i)3-s + (−0.987 − 0.154i)4-s + (0.144 + 0.0108i)5-s + (0.578 − 0.721i)6-s + (−0.484 − 0.874i)7-s + (0.231 − 0.972i)8-s + (−0.0530 − 0.135i)9-s + (−0.0220 + 0.143i)10-s + (−0.787 − 0.309i)11-s + (0.674 + 0.632i)12-s + (0.558 − 0.445i)13-s + (0.909 − 0.415i)14-s + (−0.104 − 0.0836i)15-s + (0.952 + 0.306i)16-s + (−0.0715 − 0.0771i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.376 + 0.926i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.376 + 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.459303 - 0.309221i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.459303 - 0.309221i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.109 - 1.40i)T \) |
| 7 | \( 1 + (1.28 + 2.31i)T \) |
| good | 3 | \( 1 + (1.32 + 0.902i)T + (1.09 + 2.79i)T^{2} \) |
| 5 | \( 1 + (-0.323 - 0.0242i)T + (4.94 + 0.745i)T^{2} \) |
| 11 | \( 1 + (2.61 + 1.02i)T + (8.06 + 7.48i)T^{2} \) |
| 13 | \( 1 + (-2.01 + 1.60i)T + (2.89 - 12.6i)T^{2} \) |
| 17 | \( 1 + (0.295 + 0.317i)T + (-1.27 + 16.9i)T^{2} \) |
| 19 | \( 1 + (-3.19 + 5.53i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (3.32 - 3.57i)T + (-1.71 - 22.9i)T^{2} \) |
| 29 | \( 1 + (0.987 + 4.32i)T + (-26.1 + 12.5i)T^{2} \) |
| 31 | \( 1 + (-1.28 - 2.21i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (8.40 + 2.59i)T + (30.5 + 20.8i)T^{2} \) |
| 41 | \( 1 + (-3.27 - 6.81i)T + (-25.5 + 32.0i)T^{2} \) |
| 43 | \( 1 + (-2.17 + 4.51i)T + (-26.8 - 33.6i)T^{2} \) |
| 47 | \( 1 + (-5.45 + 0.822i)T + (44.9 - 13.8i)T^{2} \) |
| 53 | \( 1 + (-5.41 + 1.67i)T + (43.7 - 29.8i)T^{2} \) |
| 59 | \( 1 + (0.182 + 2.43i)T + (-58.3 + 8.79i)T^{2} \) |
| 61 | \( 1 + (-2.02 + 6.56i)T + (-50.4 - 34.3i)T^{2} \) |
| 67 | \( 1 + (0.882 - 0.509i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-11.9 - 2.72i)T + (63.9 + 30.8i)T^{2} \) |
| 73 | \( 1 + (-1.03 + 6.87i)T + (-69.7 - 21.5i)T^{2} \) |
| 79 | \( 1 + (-9.00 - 5.19i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (3.97 - 4.98i)T + (-18.4 - 80.9i)T^{2} \) |
| 89 | \( 1 + (-12.6 + 4.97i)T + (65.2 - 60.5i)T^{2} \) |
| 97 | \( 1 - 8.05iT - 97T^{2} \) |
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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
−12.57477716687352762097725142627, −11.32711808276282735411572605739, −10.26126610296712300129148275838, −9.264220707103871946221443024761, −7.922315024701139430931807337544, −7.06626355285178721850391004336, −6.12301278745439781241523422618, −5.25531847030041239518459275319, −3.64292591229047396780028824111, −0.54695755831253906446129458373,
2.23011968137311447603765209110, 3.82253340689641632827985741611, 5.19685718141102176480741667525, 5.95062379737658694388558797938, 7.921689841379976621584942322697, 9.028656881870037150418585537640, 10.06290321873168944416410655468, 10.63542861625038681750866567336, 11.77621419242094098908388755502, 12.30464316163758689417365334010
