L(s) = 1 | + (−1.36 − 0.351i)2-s + (−1.52 − 0.712i)3-s + (1.75 + 0.961i)4-s + (2.13 + 3.04i)5-s + (1.84 + 1.51i)6-s + (0.591 + 1.02i)7-s + (−2.06 − 1.93i)8-s + (−0.0997 − 0.118i)9-s + (−1.85 − 4.91i)10-s + (−6.12 + 1.64i)11-s + (−1.99 − 2.72i)12-s + (0.768 − 0.358i)13-s + (−0.450 − 1.61i)14-s + (−1.08 − 6.17i)15-s + (2.14 + 3.37i)16-s + (0.0293 + 0.0246i)17-s + ⋯ |
L(s) = 1 | + (−0.968 − 0.248i)2-s + (−0.882 − 0.411i)3-s + (0.876 + 0.480i)4-s + (0.952 + 1.36i)5-s + (0.752 + 0.617i)6-s + (0.223 + 0.387i)7-s + (−0.729 − 0.683i)8-s + (−0.0332 − 0.0396i)9-s + (−0.585 − 1.55i)10-s + (−1.84 + 0.494i)11-s + (−0.575 − 0.785i)12-s + (0.213 − 0.0993i)13-s + (−0.120 − 0.430i)14-s + (−0.280 − 1.59i)15-s + (0.537 + 0.843i)16-s + (0.00711 + 0.00596i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.254 - 0.966i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.254 - 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.282424 + 0.366494i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.282424 + 0.366494i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.36 + 0.351i)T \) |
| 19 | \( 1 + (3.35 - 2.78i)T \) |
good | 3 | \( 1 + (1.52 + 0.712i)T + (1.92 + 2.29i)T^{2} \) |
| 5 | \( 1 + (-2.13 - 3.04i)T + (-1.71 + 4.69i)T^{2} \) |
| 7 | \( 1 + (-0.591 - 1.02i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (6.12 - 1.64i)T + (9.52 - 5.5i)T^{2} \) |
| 13 | \( 1 + (-0.768 + 0.358i)T + (8.35 - 9.95i)T^{2} \) |
| 17 | \( 1 + (-0.0293 - 0.0246i)T + (2.95 + 16.7i)T^{2} \) |
| 23 | \( 1 + (-0.511 - 2.90i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (0.147 - 1.68i)T + (-28.5 - 5.03i)T^{2} \) |
| 31 | \( 1 + (2.04 + 3.53i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-4.10 - 4.10i)T + 37iT^{2} \) |
| 41 | \( 1 + (11.1 - 4.05i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (-5.24 + 3.67i)T + (14.7 - 40.4i)T^{2} \) |
| 47 | \( 1 + (-2.81 - 3.35i)T + (-8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + (3.60 - 5.15i)T + (-18.1 - 49.8i)T^{2} \) |
| 59 | \( 1 + (1.05 - 0.0920i)T + (58.1 - 10.2i)T^{2} \) |
| 61 | \( 1 + (7.22 - 10.3i)T + (-20.8 - 57.3i)T^{2} \) |
| 67 | \( 1 + (-0.417 - 0.0364i)T + (65.9 + 11.6i)T^{2} \) |
| 71 | \( 1 + (-11.8 - 2.09i)T + (66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (1.51 + 4.16i)T + (-55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (-10.3 + 3.77i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (-11.1 - 2.99i)T + (71.8 + 41.5i)T^{2} \) |
| 89 | \( 1 + (5.37 + 1.95i)T + (68.1 + 57.2i)T^{2} \) |
| 97 | \( 1 + (-8.28 + 9.87i)T + (-16.8 - 95.5i)T^{2} \) |
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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
−11.70878013042279703761888826252, −10.72974020353293137185413994142, −10.45445297217196331184675767209, −9.430233011401666970713736707184, −8.061124906087305807677211478572, −7.14836084325586843596767875824, −6.22511159800383423229046144572, −5.50011252072762915121807438355, −3.02199607927282901616492203948, −1.99534163868324915095643466408,
0.47612557140666620014600794422, 2.26668429144273972079374714487, 4.88359438245629292110922948224, 5.41113358964253101674457903385, 6.36315238689458903216963434272, 7.917258976480960218575417030338, 8.618689319247546409901875752131, 9.604621756575368699319928575857, 10.65460024203125099331352032897, 10.87477965821479549282042435549