| L(s) = 1 | + (0.669 + 0.743i)2-s + (0.913 + 0.406i)3-s + (−0.104 + 0.994i)4-s + (3.97 + 0.845i)5-s + (0.309 + 0.951i)6-s + (−2.27 − 1.34i)7-s + (−0.809 + 0.587i)8-s + (0.669 + 0.743i)9-s + (2.03 + 3.51i)10-s + (0.0572 + 3.31i)11-s + (−0.5 + 0.866i)12-s + (0.914 − 2.81i)13-s + (−0.525 − 2.59i)14-s + (3.28 + 2.38i)15-s + (−0.978 − 0.207i)16-s + (−2.99 + 3.32i)17-s + ⋯ |
| L(s) = 1 | + (0.473 + 0.525i)2-s + (0.527 + 0.234i)3-s + (−0.0522 + 0.497i)4-s + (1.77 + 0.377i)5-s + (0.126 + 0.388i)6-s + (−0.861 − 0.508i)7-s + (−0.286 + 0.207i)8-s + (0.223 + 0.247i)9-s + (0.642 + 1.11i)10-s + (0.0172 + 0.999i)11-s + (−0.144 + 0.249i)12-s + (0.253 − 0.780i)13-s + (−0.140 − 0.693i)14-s + (0.848 + 0.616i)15-s + (−0.244 − 0.0519i)16-s + (−0.725 + 0.805i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.395 - 0.918i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.395 - 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.07722 + 1.36688i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.07722 + 1.36688i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.669 - 0.743i)T \) |
| 3 | \( 1 + (-0.913 - 0.406i)T \) |
| 7 | \( 1 + (2.27 + 1.34i)T \) |
| 11 | \( 1 + (-0.0572 - 3.31i)T \) |
| good | 5 | \( 1 + (-3.97 - 0.845i)T + (4.56 + 2.03i)T^{2} \) |
| 13 | \( 1 + (-0.914 + 2.81i)T + (-10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (2.99 - 3.32i)T + (-1.77 - 16.9i)T^{2} \) |
| 19 | \( 1 + (0.764 + 7.27i)T + (-18.5 + 3.95i)T^{2} \) |
| 23 | \( 1 + (-1.81 + 3.13i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (0.324 + 0.235i)T + (8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (5.98 - 1.27i)T + (28.3 - 12.6i)T^{2} \) |
| 37 | \( 1 + (7.90 - 3.52i)T + (24.7 - 27.4i)T^{2} \) |
| 41 | \( 1 + (-5.50 + 3.99i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + 1.88T + 43T^{2} \) |
| 47 | \( 1 + (-0.0136 - 0.129i)T + (-45.9 + 9.77i)T^{2} \) |
| 53 | \( 1 + (-0.0210 + 0.00447i)T + (48.4 - 21.5i)T^{2} \) |
| 59 | \( 1 + (-0.328 + 3.12i)T + (-57.7 - 12.2i)T^{2} \) |
| 61 | \( 1 + (-7.52 - 1.59i)T + (55.7 + 24.8i)T^{2} \) |
| 67 | \( 1 + (-2.73 - 4.73i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (4.19 + 12.9i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (0.346 - 3.29i)T + (-71.4 - 15.1i)T^{2} \) |
| 79 | \( 1 + (11.1 + 12.3i)T + (-8.25 + 78.5i)T^{2} \) |
| 83 | \( 1 + (-2.23 - 6.88i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (0.223 - 0.386i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-5.58 + 17.1i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−10.82473382134460664577661331726, −10.27023362131820696007357269716, −9.377223236611336929378938858704, −8.721087177465441298889080279912, −7.12504699976157471061740550766, −6.66154891100239000169541582949, −5.62752329797899491084354707130, −4.56718033375182387012068767984, −3.17340176832180999277605494455, −2.14704458148117902085903072658,
1.59517326599854880222977371614, 2.57996267935023421431664749365, 3.73943885356737236408845803294, 5.38227298204039699491160462776, 5.98445954725358454755542292236, 6.84942757663827473319087691294, 8.620879959762585025039219161614, 9.264167280770900715862619147671, 9.807559435554332016599725131792, 10.82080733740295654325897720965
