| L(s) = 1 | + (0.800 + 0.599i)2-s + (−1.19 + 0.447i)3-s + (0.281 + 0.959i)4-s + (−1.17 + 1.90i)5-s + (−1.22 − 0.360i)6-s + (−4.10 − 0.892i)7-s + (−0.349 + 0.936i)8-s + (−1.02 + 0.892i)9-s + (−2.08 + 0.818i)10-s + (2.30 + 0.330i)11-s + (−0.766 − 1.02i)12-s + (−0.202 − 0.932i)13-s + (−2.74 − 3.17i)14-s + (0.557 − 2.80i)15-s + (−0.841 + 0.540i)16-s + (3.53 + 6.47i)17-s + ⋯ |
| L(s) = 1 | + (0.566 + 0.423i)2-s + (−0.692 + 0.258i)3-s + (0.140 + 0.479i)4-s + (−0.525 + 0.850i)5-s + (−0.501 − 0.147i)6-s + (−1.55 − 0.337i)7-s + (−0.123 + 0.331i)8-s + (−0.343 + 0.297i)9-s + (−0.657 + 0.258i)10-s + (0.694 + 0.0997i)11-s + (−0.221 − 0.295i)12-s + (−0.0562 − 0.258i)13-s + (−0.734 − 0.847i)14-s + (0.144 − 0.724i)15-s + (−0.210 + 0.135i)16-s + (0.857 + 1.56i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.895 - 0.445i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.895 - 0.445i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.184771 + 0.785795i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.184771 + 0.785795i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.800 - 0.599i)T \) |
| 5 | \( 1 + (1.17 - 1.90i)T \) |
| 23 | \( 1 + (4.75 - 0.628i)T \) |
| good | 3 | \( 1 + (1.19 - 0.447i)T + (2.26 - 1.96i)T^{2} \) |
| 7 | \( 1 + (4.10 + 0.892i)T + (6.36 + 2.90i)T^{2} \) |
| 11 | \( 1 + (-2.30 - 0.330i)T + (10.5 + 3.09i)T^{2} \) |
| 13 | \( 1 + (0.202 + 0.932i)T + (-11.8 + 5.40i)T^{2} \) |
| 17 | \( 1 + (-3.53 - 6.47i)T + (-9.19 + 14.3i)T^{2} \) |
| 19 | \( 1 + (-2.40 + 0.706i)T + (15.9 - 10.2i)T^{2} \) |
| 29 | \( 1 + (1.30 - 4.45i)T + (-24.3 - 15.6i)T^{2} \) |
| 31 | \( 1 + (-4.15 - 9.09i)T + (-20.3 + 23.4i)T^{2} \) |
| 37 | \( 1 + (0.541 - 7.57i)T + (-36.6 - 5.26i)T^{2} \) |
| 41 | \( 1 + (-5.61 + 6.47i)T + (-5.83 - 40.5i)T^{2} \) |
| 43 | \( 1 + (2.77 + 7.43i)T + (-32.4 + 28.1i)T^{2} \) |
| 47 | \( 1 + (-7.75 + 7.75i)T - 47iT^{2} \) |
| 53 | \( 1 + (0.963 - 4.42i)T + (-48.2 - 22.0i)T^{2} \) |
| 59 | \( 1 + (-2.26 + 3.53i)T + (-24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (6.04 - 2.76i)T + (39.9 - 46.1i)T^{2} \) |
| 67 | \( 1 + (6.85 - 9.15i)T + (-18.8 - 64.2i)T^{2} \) |
| 71 | \( 1 + (-0.111 - 0.774i)T + (-68.1 + 20.0i)T^{2} \) |
| 73 | \( 1 + (-2.62 - 1.43i)T + (39.4 + 61.4i)T^{2} \) |
| 79 | \( 1 + (-0.534 - 0.343i)T + (32.8 + 71.8i)T^{2} \) |
| 83 | \( 1 + (9.40 + 0.672i)T + (82.1 + 11.8i)T^{2} \) |
| 89 | \( 1 + (-1.06 + 2.34i)T + (-58.2 - 67.2i)T^{2} \) |
| 97 | \( 1 + (0.460 - 0.0329i)T + (96.0 - 13.8i)T^{2} \) |
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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
−12.38593598854635099916153907999, −11.93261329551910424784529588101, −10.58384700345536261015967809639, −10.13591331515317927385213127985, −8.535029804427184237399209649653, −7.25404145640399869500999972744, −6.42325516750224321618630809322, −5.63449111450769324217743445423, −3.98044049032814238358870419279, −3.14811512949486847703030957772,
0.59613864139515593806353628792, 3.02038787859571899858105775647, 4.23865796289510395256874135390, 5.67157345773241428907601346546, 6.29781391476563974059482905687, 7.63001283794656012608118959949, 9.358451761389842307018999456911, 9.609507634161236504289673425362, 11.37343905874073246803575751543, 11.94664409118027505877320638874
