| L(s) = 1 | + (−1.34 + 0.433i)2-s + (−0.132 + 0.924i)3-s + (1.62 − 1.16i)4-s + (−0.959 + 0.281i)5-s + (−0.222 − 1.30i)6-s + (2.80 + 3.23i)7-s + (−1.67 + 2.27i)8-s + (2.04 + 0.599i)9-s + (1.16 − 0.795i)10-s + (−3.09 + 4.81i)11-s + (0.863 + 1.65i)12-s + (2.73 + 2.37i)13-s + (−5.18 − 3.14i)14-s + (−0.132 − 0.924i)15-s + (1.27 − 3.79i)16-s + (4.43 − 2.02i)17-s + ⋯ |
| L(s) = 1 | + (−0.951 + 0.306i)2-s + (−0.0767 + 0.533i)3-s + (0.811 − 0.583i)4-s + (−0.429 + 0.125i)5-s + (−0.0906 − 0.531i)6-s + (1.06 + 1.22i)7-s + (−0.593 + 0.804i)8-s + (0.680 + 0.199i)9-s + (0.369 − 0.251i)10-s + (−0.932 + 1.45i)11-s + (0.249 + 0.477i)12-s + (0.759 + 0.658i)13-s + (−1.38 − 0.839i)14-s + (−0.0343 − 0.238i)15-s + (0.318 − 0.948i)16-s + (1.07 − 0.491i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.725 - 0.688i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.725 - 0.688i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.375332 + 0.940931i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.375332 + 0.940931i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.34 - 0.433i)T \) |
| 5 | \( 1 + (0.959 - 0.281i)T \) |
| 23 | \( 1 + (1.78 + 4.45i)T \) |
| good | 3 | \( 1 + (0.132 - 0.924i)T + (-2.87 - 0.845i)T^{2} \) |
| 7 | \( 1 + (-2.80 - 3.23i)T + (-0.996 + 6.92i)T^{2} \) |
| 11 | \( 1 + (3.09 - 4.81i)T + (-4.56 - 10.0i)T^{2} \) |
| 13 | \( 1 + (-2.73 - 2.37i)T + (1.85 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-4.43 + 2.02i)T + (11.1 - 12.8i)T^{2} \) |
| 19 | \( 1 + (3.44 + 1.57i)T + (12.4 + 14.3i)T^{2} \) |
| 29 | \( 1 + (-2.94 + 1.34i)T + (18.9 - 21.9i)T^{2} \) |
| 31 | \( 1 + (-4.96 + 0.713i)T + (29.7 - 8.73i)T^{2} \) |
| 37 | \( 1 + (-6.20 - 1.82i)T + (31.1 + 20.0i)T^{2} \) |
| 41 | \( 1 + (11.1 - 3.28i)T + (34.4 - 22.1i)T^{2} \) |
| 43 | \( 1 + (4.00 + 0.576i)T + (41.2 + 12.1i)T^{2} \) |
| 47 | \( 1 + 2.40iT - 47T^{2} \) |
| 53 | \( 1 + (-6.44 - 7.43i)T + (-7.54 + 52.4i)T^{2} \) |
| 59 | \( 1 + (3.78 - 4.37i)T + (-8.39 - 58.3i)T^{2} \) |
| 61 | \( 1 + (0.158 + 1.10i)T + (-58.5 + 17.1i)T^{2} \) |
| 67 | \( 1 + (4.57 + 7.12i)T + (-27.8 + 60.9i)T^{2} \) |
| 71 | \( 1 + (1.73 + 2.70i)T + (-29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (-5.43 + 11.8i)T + (-47.8 - 55.1i)T^{2} \) |
| 79 | \( 1 + (10.6 - 12.2i)T + (-11.2 - 78.1i)T^{2} \) |
| 83 | \( 1 + (-1.16 + 3.97i)T + (-69.8 - 44.8i)T^{2} \) |
| 89 | \( 1 + (4.72 + 0.679i)T + (85.3 + 25.0i)T^{2} \) |
| 97 | \( 1 + (-4.85 - 16.5i)T + (-81.6 + 52.4i)T^{2} \) |
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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
−10.24986273521729959405950806319, −9.629417297420329417845319731266, −8.603055453310806708182395234897, −8.056446000567204069847876652574, −7.23880603217556570094306812491, −6.23490866502440161866859037326, −5.03941744207574952956756630547, −4.52341242683537637553518480898, −2.63695700223907320157042971207, −1.67691389672579293128365183330,
0.72611492881328443549999016310, 1.51861527859336180334816416563, 3.25385332675360778251489728795, 4.06763458437788686314586324065, 5.57639603267676596272486921694, 6.61898992745078707390686793268, 7.64193967704727792762084597457, 8.073763101018759677419799418935, 8.523696843113463777273954197441, 10.24014265974807380526771264529
