| L(s) = 1 | + (−1.16 − 0.800i)2-s + (0.338 + 3.00i)3-s + (0.716 + 1.86i)4-s + (1.55 + 0.748i)5-s + (2.01 − 3.77i)6-s + (2.09 + 1.66i)7-s + (0.660 − 2.75i)8-s + (−5.99 + 1.36i)9-s + (−1.21 − 2.11i)10-s + (−0.335 − 0.534i)11-s + (−5.36 + 2.78i)12-s + (1.04 − 4.58i)13-s + (−1.10 − 3.61i)14-s + (−1.72 + 4.92i)15-s + (−2.97 + 2.67i)16-s + (−5.02 + 5.02i)17-s + ⋯ |
| L(s) = 1 | + (−0.824 − 0.566i)2-s + (0.195 + 1.73i)3-s + (0.358 + 0.933i)4-s + (0.695 + 0.334i)5-s + (0.821 − 1.54i)6-s + (0.790 + 0.630i)7-s + (0.233 − 0.972i)8-s + (−1.99 + 0.455i)9-s + (−0.383 − 0.669i)10-s + (−0.101 − 0.161i)11-s + (−1.54 + 0.804i)12-s + (0.290 − 1.27i)13-s + (−0.294 − 0.966i)14-s + (−0.445 + 1.27i)15-s + (−0.743 + 0.669i)16-s + (−1.21 + 1.21i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0480 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0480 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.736733 + 0.702151i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.736733 + 0.702151i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.16 + 0.800i)T \) |
| 29 | \( 1 + (-1.22 + 5.24i)T \) |
| good | 3 | \( 1 + (-0.338 - 3.00i)T + (-2.92 + 0.667i)T^{2} \) |
| 5 | \( 1 + (-1.55 - 0.748i)T + (3.11 + 3.90i)T^{2} \) |
| 7 | \( 1 + (-2.09 - 1.66i)T + (1.55 + 6.82i)T^{2} \) |
| 11 | \( 1 + (0.335 + 0.534i)T + (-4.77 + 9.91i)T^{2} \) |
| 13 | \( 1 + (-1.04 + 4.58i)T + (-11.7 - 5.64i)T^{2} \) |
| 17 | \( 1 + (5.02 - 5.02i)T - 17iT^{2} \) |
| 19 | \( 1 + (-5.95 - 0.671i)T + (18.5 + 4.22i)T^{2} \) |
| 23 | \( 1 + (1.96 + 4.07i)T + (-14.3 + 17.9i)T^{2} \) |
| 31 | \( 1 + (-2.48 - 7.09i)T + (-24.2 + 19.3i)T^{2} \) |
| 37 | \( 1 + (-0.312 - 0.196i)T + (16.0 + 33.3i)T^{2} \) |
| 41 | \( 1 + (-0.0616 - 0.0616i)T + 41iT^{2} \) |
| 43 | \( 1 + (0.767 - 2.19i)T + (-33.6 - 26.8i)T^{2} \) |
| 47 | \( 1 + (4.14 + 6.60i)T + (-20.3 + 42.3i)T^{2} \) |
| 53 | \( 1 + (-0.00838 + 0.0174i)T + (-33.0 - 41.4i)T^{2} \) |
| 59 | \( 1 - 8.07T + 59T^{2} \) |
| 61 | \( 1 + (2.62 - 0.296i)T + (59.4 - 13.5i)T^{2} \) |
| 67 | \( 1 + (0.971 - 0.221i)T + (60.3 - 29.0i)T^{2} \) |
| 71 | \( 1 + (-1.61 + 7.06i)T + (-63.9 - 30.8i)T^{2} \) |
| 73 | \( 1 + (-10.6 - 3.71i)T + (57.0 + 45.5i)T^{2} \) |
| 79 | \( 1 + (0.163 - 0.260i)T + (-34.2 - 71.1i)T^{2} \) |
| 83 | \( 1 + (-2.45 - 3.07i)T + (-18.4 + 80.9i)T^{2} \) |
| 89 | \( 1 + (-5.29 + 1.85i)T + (69.5 - 55.4i)T^{2} \) |
| 97 | \( 1 + (0.0988 - 0.877i)T + (-94.5 - 21.5i)T^{2} \) |
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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
−11.97506868056184993882987568330, −11.00665351923015177990896436117, −10.36317539634399118332817760761, −9.789067750065366198473858911696, −8.640730786912454205828342685805, −8.181026121246212373056397035813, −6.13931562307420927470487877949, −4.92845991746383410490334107549, −3.58136310872030286340217565364, −2.39224383284267707905950780614,
1.19896566590488242439870933745, 2.16364477432185419202812024423, 5.02837087831406712320148952926, 6.26033760653523708395884594043, 7.17195920425148375422413538722, 7.71378125828751437346357052841, 8.885709848231243732012735249525, 9.600840856812941411780352043849, 11.31466857987069044736530214705, 11.66271885899159390539820678471
