| L(s) = 1 | + (0.393 + 0.121i)2-s + (−1.51 − 1.03i)4-s + (1.73 − 0.260i)5-s + (−2.49 + 0.867i)7-s + (−0.982 − 1.23i)8-s + (0.712 + 0.107i)10-s + (3.93 − 3.64i)11-s + (−1.06 − 4.66i)13-s + (−1.08 + 0.0377i)14-s + (1.10 + 2.80i)16-s + (−0.0319 − 0.426i)17-s + (2.07 − 3.59i)19-s + (−2.88 − 1.39i)20-s + (1.98 − 0.957i)22-s + (0.321 − 4.28i)23-s + ⋯ |
| L(s) = 1 | + (0.278 + 0.0857i)2-s + (−0.756 − 0.515i)4-s + (0.774 − 0.116i)5-s + (−0.944 + 0.327i)7-s + (−0.347 − 0.435i)8-s + (0.225 + 0.0339i)10-s + (1.18 − 1.09i)11-s + (−0.295 − 1.29i)13-s + (−0.290 + 0.0100i)14-s + (0.275 + 0.701i)16-s + (−0.00774 − 0.103i)17-s + (0.476 − 0.825i)19-s + (−0.645 − 0.310i)20-s + (0.423 − 0.204i)22-s + (0.0670 − 0.894i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.185 + 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.185 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.988502 - 0.819038i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.988502 - 0.819038i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + (2.49 - 0.867i)T \) |
| good | 2 | \( 1 + (-0.393 - 0.121i)T + (1.65 + 1.12i)T^{2} \) |
| 5 | \( 1 + (-1.73 + 0.260i)T + (4.77 - 1.47i)T^{2} \) |
| 11 | \( 1 + (-3.93 + 3.64i)T + (0.822 - 10.9i)T^{2} \) |
| 13 | \( 1 + (1.06 + 4.66i)T + (-11.7 + 5.64i)T^{2} \) |
| 17 | \( 1 + (0.0319 + 0.426i)T + (-16.8 + 2.53i)T^{2} \) |
| 19 | \( 1 + (-2.07 + 3.59i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.321 + 4.28i)T + (-22.7 - 3.42i)T^{2} \) |
| 29 | \( 1 + (3.37 + 1.62i)T + (18.0 + 22.6i)T^{2} \) |
| 31 | \( 1 + (-0.751 - 1.30i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (1.35 - 0.922i)T + (13.5 - 34.4i)T^{2} \) |
| 41 | \( 1 + (2.12 + 2.66i)T + (-9.12 + 39.9i)T^{2} \) |
| 43 | \( 1 + (0.263 - 0.330i)T + (-9.56 - 41.9i)T^{2} \) |
| 47 | \( 1 + (-8.66 - 2.67i)T + (38.8 + 26.4i)T^{2} \) |
| 53 | \( 1 + (-11.4 - 7.78i)T + (19.3 + 49.3i)T^{2} \) |
| 59 | \( 1 + (0.853 + 0.128i)T + (56.3 + 17.3i)T^{2} \) |
| 61 | \( 1 + (4.20 - 2.86i)T + (22.2 - 56.7i)T^{2} \) |
| 67 | \( 1 + (-7.14 - 12.3i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-12.3 + 5.94i)T + (44.2 - 55.5i)T^{2} \) |
| 73 | \( 1 + (15.1 - 4.67i)T + (60.3 - 41.1i)T^{2} \) |
| 79 | \( 1 + (2.68 - 4.64i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (1.23 - 5.41i)T + (-74.7 - 36.0i)T^{2} \) |
| 89 | \( 1 + (3.63 + 3.37i)T + (6.65 + 88.7i)T^{2} \) |
| 97 | \( 1 + 10.8T + 97T^{2} \) |
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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
−10.72100766821914878010975660141, −9.881450060045318586799250679692, −9.192865324063645270714582073945, −8.548567587132798498329382249672, −6.91947615576464527472625168945, −5.88415899441814729069086350769, −5.49739916585999036487705374200, −4.01543708750100744329638474895, −2.86149512801210326467143280811, −0.78617200267594130810409544946,
1.91826742264113479189379224098, 3.56713389284948699942678971960, 4.29683960279516028357722513406, 5.60002534033185380419851245116, 6.67549790125994412861115048546, 7.47154566268498529814370515965, 8.959191454188403153766166248519, 9.554951339359429083570012915104, 10.02792573491607259948561120300, 11.64042930244040065905985452399
