| L(s) = 1 | + (1.40 + 0.151i)2-s + (−0.128 − 0.0619i)3-s + (1.95 + 0.426i)4-s + (0.427 − 0.0975i)5-s + (−0.171 − 0.106i)6-s + (1.61 + 0.778i)7-s + (2.68 + 0.895i)8-s + (−1.85 − 2.32i)9-s + (0.615 − 0.0723i)10-s + (−1.88 + 2.36i)11-s + (−0.224 − 0.175i)12-s + (−3.63 − 2.89i)13-s + (2.15 + 1.33i)14-s + (−0.0610 − 0.0139i)15-s + (3.63 + 1.66i)16-s + 5.73i·17-s + ⋯ |
| L(s) = 1 | + (0.994 + 0.107i)2-s + (−0.0742 − 0.0357i)3-s + (0.977 + 0.213i)4-s + (0.191 − 0.0436i)5-s + (−0.0700 − 0.0435i)6-s + (0.610 + 0.294i)7-s + (0.948 + 0.316i)8-s + (−0.619 − 0.776i)9-s + (0.194 − 0.0228i)10-s + (−0.569 + 0.713i)11-s + (−0.0649 − 0.0507i)12-s + (−1.00 − 0.802i)13-s + (0.575 + 0.357i)14-s + (−0.0157 − 0.00359i)15-s + (0.909 + 0.416i)16-s + 1.39i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 - 0.162i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.986 - 0.162i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.15096 + 0.176007i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.15096 + 0.176007i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.40 - 0.151i)T \) |
| 29 | \( 1 + (5.11 + 1.69i)T \) |
| good | 3 | \( 1 + (0.128 + 0.0619i)T + (1.87 + 2.34i)T^{2} \) |
| 5 | \( 1 + (-0.427 + 0.0975i)T + (4.50 - 2.16i)T^{2} \) |
| 7 | \( 1 + (-1.61 - 0.778i)T + (4.36 + 5.47i)T^{2} \) |
| 11 | \( 1 + (1.88 - 2.36i)T + (-2.44 - 10.7i)T^{2} \) |
| 13 | \( 1 + (3.63 + 2.89i)T + (2.89 + 12.6i)T^{2} \) |
| 17 | \( 1 - 5.73iT - 17T^{2} \) |
| 19 | \( 1 + (-6.82 + 3.28i)T + (11.8 - 14.8i)T^{2} \) |
| 23 | \( 1 + (-0.950 + 4.16i)T + (-20.7 - 9.97i)T^{2} \) |
| 31 | \( 1 + (8.43 - 1.92i)T + (27.9 - 13.4i)T^{2} \) |
| 37 | \( 1 + (7.07 + 8.87i)T + (-8.23 + 36.0i)T^{2} \) |
| 41 | \( 1 - 1.02iT - 41T^{2} \) |
| 43 | \( 1 + (1.50 - 6.60i)T + (-38.7 - 18.6i)T^{2} \) |
| 47 | \( 1 + (-5.07 - 4.04i)T + (10.4 + 45.8i)T^{2} \) |
| 53 | \( 1 + (-4.09 + 0.934i)T + (47.7 - 22.9i)T^{2} \) |
| 59 | \( 1 + 0.924iT - 59T^{2} \) |
| 61 | \( 1 + (-7.25 - 3.49i)T + (38.0 + 47.6i)T^{2} \) |
| 67 | \( 1 + (-1.78 + 1.42i)T + (14.9 - 65.3i)T^{2} \) |
| 71 | \( 1 + (-1.73 + 2.18i)T + (-15.7 - 69.2i)T^{2} \) |
| 73 | \( 1 + (-5.05 - 1.15i)T + (65.7 + 31.6i)T^{2} \) |
| 79 | \( 1 + (-7.30 + 5.82i)T + (17.5 - 77.0i)T^{2} \) |
| 83 | \( 1 + (-7.72 - 16.0i)T + (-51.7 + 64.8i)T^{2} \) |
| 89 | \( 1 + (-12.3 + 2.82i)T + (80.1 - 38.6i)T^{2} \) |
| 97 | \( 1 + (3.37 + 7.00i)T + (-60.4 + 75.8i)T^{2} \) |
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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
−12.42344883870679282202091349692, −11.48124267756513181844188843020, −10.56865411443149886791141827199, −9.359464549726061928480129909040, −7.960001770989529577684275844062, −7.11892224015367465590771633387, −5.69412957033254858880253121934, −5.11966859138216964886023644817, −3.58297824402540801040095481653, −2.19587672158509217192972526285,
2.09304690104451857326335609828, 3.47497513426422852539337970689, 5.12441000602213854886366929279, 5.45983915034758079061281014625, 7.18087178288048276090871586444, 7.81323636310482164432107670888, 9.451593480723226139869807202086, 10.53692483139703592347116453591, 11.54477532471534222753824754328, 11.89147270181401556544931550697
