| L(s) = 1 | + (0.688 + 1.23i)2-s + (−0.129 + 1.31i)3-s + (−1.05 + 1.70i)4-s + (−3.05 + 1.63i)5-s + (−1.71 + 0.746i)6-s + (0.195 + 0.980i)7-s + (−2.82 − 0.129i)8-s + (1.22 + 0.243i)9-s + (−4.12 − 2.65i)10-s + (−0.963 + 1.17i)11-s + (−2.10 − 1.60i)12-s + (−1.86 + 3.49i)13-s + (−1.07 + 0.916i)14-s + (−1.75 − 4.23i)15-s + (−1.78 − 3.57i)16-s + (1.60 − 3.87i)17-s + ⋯ |
| L(s) = 1 | + (0.486 + 0.873i)2-s + (−0.0748 + 0.760i)3-s + (−0.526 + 0.850i)4-s + (−1.36 + 0.730i)5-s + (−0.700 + 0.304i)6-s + (0.0737 + 0.370i)7-s + (−0.998 − 0.0456i)8-s + (0.408 + 0.0812i)9-s + (−1.30 − 0.838i)10-s + (−0.290 + 0.353i)11-s + (−0.606 − 0.463i)12-s + (−0.518 + 0.969i)13-s + (−0.287 + 0.244i)14-s + (−0.453 − 1.09i)15-s + (−0.446 − 0.894i)16-s + (0.389 − 0.940i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.219 + 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.219 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.569681 - 0.712024i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.569681 - 0.712024i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.688 - 1.23i)T \) |
| 7 | \( 1 + (-0.195 - 0.980i)T \) |
| good | 3 | \( 1 + (0.129 - 1.31i)T + (-2.94 - 0.585i)T^{2} \) |
| 5 | \( 1 + (3.05 - 1.63i)T + (2.77 - 4.15i)T^{2} \) |
| 11 | \( 1 + (0.963 - 1.17i)T + (-2.14 - 10.7i)T^{2} \) |
| 13 | \( 1 + (1.86 - 3.49i)T + (-7.22 - 10.8i)T^{2} \) |
| 17 | \( 1 + (-1.60 + 3.87i)T + (-12.0 - 12.0i)T^{2} \) |
| 19 | \( 1 + (-0.345 - 1.13i)T + (-15.7 + 10.5i)T^{2} \) |
| 23 | \( 1 + (-4.01 + 2.68i)T + (8.80 - 21.2i)T^{2} \) |
| 29 | \( 1 + (-0.526 + 0.432i)T + (5.65 - 28.4i)T^{2} \) |
| 31 | \( 1 + (1.11 - 1.11i)T - 31iT^{2} \) |
| 37 | \( 1 + (4.15 + 1.26i)T + (30.7 + 20.5i)T^{2} \) |
| 41 | \( 1 + (-3.74 - 5.60i)T + (-15.6 + 37.8i)T^{2} \) |
| 43 | \( 1 + (0.289 + 2.94i)T + (-42.1 + 8.38i)T^{2} \) |
| 47 | \( 1 + (1.19 + 0.492i)T + (33.2 + 33.2i)T^{2} \) |
| 53 | \( 1 + (1.06 + 0.874i)T + (10.3 + 51.9i)T^{2} \) |
| 59 | \( 1 + (0.510 + 0.954i)T + (-32.7 + 49.0i)T^{2} \) |
| 61 | \( 1 + (9.27 + 0.913i)T + (59.8 + 11.9i)T^{2} \) |
| 67 | \( 1 + (-4.43 - 0.437i)T + (65.7 + 13.0i)T^{2} \) |
| 71 | \( 1 + (-3.56 + 0.709i)T + (65.5 - 27.1i)T^{2} \) |
| 73 | \( 1 + (1.87 - 9.42i)T + (-67.4 - 27.9i)T^{2} \) |
| 79 | \( 1 + (11.4 - 4.73i)T + (55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + (-12.2 + 3.71i)T + (69.0 - 46.1i)T^{2} \) |
| 89 | \( 1 + (11.8 + 7.90i)T + (34.0 + 82.2i)T^{2} \) |
| 97 | \( 1 + (5.53 - 5.53i)T - 97iT^{2} \) |
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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
−10.78388945903211506572805148309, −9.731448331135054485339582904983, −8.977700907859419059231939579020, −7.912992786622630816334583992820, −7.24281334918318807772477686398, −6.67718253567185472512089052456, −5.20899775680977613347054990176, −4.55909354303308700950257804012, −3.77805423552118125652388291961, −2.79407143359020943287838780868,
0.40187918798906891646491417807, 1.43404103888848003464913496707, 3.07582922755866801545165303175, 3.98000099349983023063866929805, 4.84551791975610719072474129462, 5.79956095128693605849830181806, 7.10530433345318291656649936164, 7.85187461287331147530929738960, 8.571633766373361693226011003951, 9.671528681017391446942207473225
