| L(s) = 1 | + (0.893 − 0.448i)2-s + (−1.57 + 0.714i)3-s + (0.597 − 0.802i)4-s + (0.891 − 2.97i)5-s + (−1.08 + 1.34i)6-s + (0.275 − 0.638i)7-s + (0.173 − 0.984i)8-s + (1.97 − 2.25i)9-s + (−0.539 − 3.06i)10-s + (5.37 + 1.27i)11-s + (−0.368 + 1.69i)12-s + (−2.73 + 1.79i)13-s + (−0.0404 − 0.693i)14-s + (0.721 + 5.33i)15-s + (−0.286 − 0.957i)16-s + (−4.35 − 1.58i)17-s + ⋯ |
| L(s) = 1 | + (0.631 − 0.317i)2-s + (−0.910 + 0.412i)3-s + (0.298 − 0.401i)4-s + (0.398 − 1.33i)5-s + (−0.444 + 0.549i)6-s + (0.104 − 0.241i)7-s + (0.0613 − 0.348i)8-s + (0.659 − 0.751i)9-s + (−0.170 − 0.967i)10-s + (1.62 + 0.384i)11-s + (−0.106 + 0.488i)12-s + (−0.757 + 0.498i)13-s + (−0.0107 − 0.185i)14-s + (0.186 + 1.37i)15-s + (−0.0717 − 0.239i)16-s + (−1.05 − 0.384i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.593 + 0.805i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.593 + 0.805i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.18277 - 0.597777i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.18277 - 0.597777i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.893 + 0.448i)T \) |
| 3 | \( 1 + (1.57 - 0.714i)T \) |
| good | 5 | \( 1 + (-0.891 + 2.97i)T + (-4.17 - 2.74i)T^{2} \) |
| 7 | \( 1 + (-0.275 + 0.638i)T + (-4.80 - 5.09i)T^{2} \) |
| 11 | \( 1 + (-5.37 - 1.27i)T + (9.82 + 4.93i)T^{2} \) |
| 13 | \( 1 + (2.73 - 1.79i)T + (5.14 - 11.9i)T^{2} \) |
| 17 | \( 1 + (4.35 + 1.58i)T + (13.0 + 10.9i)T^{2} \) |
| 19 | \( 1 + (2.23 - 0.813i)T + (14.5 - 12.2i)T^{2} \) |
| 23 | \( 1 + (-2.79 - 6.47i)T + (-15.7 + 16.7i)T^{2} \) |
| 29 | \( 1 + (-0.0802 + 1.37i)T + (-28.8 - 3.36i)T^{2} \) |
| 31 | \( 1 + (3.53 + 0.412i)T + (30.1 + 7.14i)T^{2} \) |
| 37 | \( 1 + (0.935 - 0.785i)T + (6.42 - 36.4i)T^{2} \) |
| 41 | \( 1 + (-10.1 - 5.09i)T + (24.4 + 32.8i)T^{2} \) |
| 43 | \( 1 + (6.37 - 6.76i)T + (-2.50 - 42.9i)T^{2} \) |
| 47 | \( 1 + (-10.4 + 1.21i)T + (45.7 - 10.8i)T^{2} \) |
| 53 | \( 1 + (2.11 + 3.66i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (5.60 - 1.32i)T + (52.7 - 26.4i)T^{2} \) |
| 61 | \( 1 + (-1.03 - 1.38i)T + (-17.4 + 58.4i)T^{2} \) |
| 67 | \( 1 + (0.0594 + 1.02i)T + (-66.5 + 7.77i)T^{2} \) |
| 71 | \( 1 + (2.04 + 11.6i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (-1.49 + 8.46i)T + (-68.5 - 24.9i)T^{2} \) |
| 79 | \( 1 + (7.86 - 3.94i)T + (47.1 - 63.3i)T^{2} \) |
| 83 | \( 1 + (8.11 - 4.07i)T + (49.5 - 66.5i)T^{2} \) |
| 89 | \( 1 + (0.642 - 3.64i)T + (-83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (-0.260 - 0.871i)T + (-81.0 + 53.3i)T^{2} \) |
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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
−12.55204893345590904782802790264, −11.84462005148970008890702973021, −11.04058002505688124959898262243, −9.536729812510434187532794496070, −9.190598682414070941361578112530, −7.08160642508245137168627241100, −5.97551388578379152140166185527, −4.77464051846751993426533248379, −4.16415855852344437063442981420, −1.45462424636730809525512148615,
2.40213236492239001614897502792, 4.19515882589399056004505086659, 5.71160675986934105377712409315, 6.62329568264148304565628629688, 7.13019547541616001189107048571, 8.850195769590307936072816922336, 10.47613444300789712951973399632, 11.08263928534936261063704332930, 12.08097173291759542158437975182, 12.94089086641302086157430787164
