| L(s) = 1 | + (0.898 + 1.09i)2-s + (0.980 + 0.195i)3-s + (−0.385 + 1.96i)4-s + (0.988 − 1.47i)5-s + (0.668 + 1.24i)6-s + (0.0771 − 0.0319i)7-s + (−2.48 + 1.34i)8-s + (0.923 + 0.382i)9-s + (2.50 − 0.249i)10-s + (0.319 + 1.60i)11-s + (−0.760 + 1.84i)12-s + (−1.00 − 1.50i)13-s + (0.104 + 0.0555i)14-s + (1.25 − 1.25i)15-s + (−3.70 − 1.51i)16-s + (−3.24 − 3.24i)17-s + ⋯ |
| L(s) = 1 | + (0.635 + 0.772i)2-s + (0.566 + 0.112i)3-s + (−0.192 + 0.981i)4-s + (0.442 − 0.661i)5-s + (0.272 + 0.508i)6-s + (0.0291 − 0.0120i)7-s + (−0.880 + 0.474i)8-s + (0.307 + 0.127i)9-s + (0.791 − 0.0790i)10-s + (0.0962 + 0.483i)11-s + (−0.219 + 0.533i)12-s + (−0.278 − 0.416i)13-s + (0.0278 + 0.0148i)14-s + (0.324 − 0.324i)15-s + (−0.925 − 0.377i)16-s + (−0.788 − 0.788i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.464 - 0.885i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.464 - 0.885i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.60228 + 0.968865i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.60228 + 0.968865i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.898 - 1.09i)T \) |
| 3 | \( 1 + (-0.980 - 0.195i)T \) |
| good | 5 | \( 1 + (-0.988 + 1.47i)T + (-1.91 - 4.61i)T^{2} \) |
| 7 | \( 1 + (-0.0771 + 0.0319i)T + (4.94 - 4.94i)T^{2} \) |
| 11 | \( 1 + (-0.319 - 1.60i)T + (-10.1 + 4.20i)T^{2} \) |
| 13 | \( 1 + (1.00 + 1.50i)T + (-4.97 + 12.0i)T^{2} \) |
| 17 | \( 1 + (3.24 + 3.24i)T + 17iT^{2} \) |
| 19 | \( 1 + (3.79 - 2.53i)T + (7.27 - 17.5i)T^{2} \) |
| 23 | \( 1 + (-2.39 + 5.79i)T + (-16.2 - 16.2i)T^{2} \) |
| 29 | \( 1 + (-1.15 + 5.82i)T + (-26.7 - 11.0i)T^{2} \) |
| 31 | \( 1 - 5.85iT - 31T^{2} \) |
| 37 | \( 1 + (3.44 + 2.30i)T + (14.1 + 34.1i)T^{2} \) |
| 41 | \( 1 + (-2.13 + 5.15i)T + (-28.9 - 28.9i)T^{2} \) |
| 43 | \( 1 + (-3.97 + 0.791i)T + (39.7 - 16.4i)T^{2} \) |
| 47 | \( 1 + (-7.75 - 7.75i)T + 47iT^{2} \) |
| 53 | \( 1 + (1.74 + 8.77i)T + (-48.9 + 20.2i)T^{2} \) |
| 59 | \( 1 + (6.80 - 10.1i)T + (-22.5 - 54.5i)T^{2} \) |
| 61 | \( 1 + (1.16 + 0.232i)T + (56.3 + 23.3i)T^{2} \) |
| 67 | \( 1 + (6.59 + 1.31i)T + (61.8 + 25.6i)T^{2} \) |
| 71 | \( 1 + (-2.11 + 0.874i)T + (50.2 - 50.2i)T^{2} \) |
| 73 | \( 1 + (-10.8 - 4.49i)T + (51.6 + 51.6i)T^{2} \) |
| 79 | \( 1 + (0.0685 - 0.0685i)T - 79iT^{2} \) |
| 83 | \( 1 + (7.79 - 5.21i)T + (31.7 - 76.6i)T^{2} \) |
| 89 | \( 1 + (-1.01 - 2.44i)T + (-62.9 + 62.9i)T^{2} \) |
| 97 | \( 1 - 16.9iT - 97T^{2} \) |
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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
−12.81001104782942452871485866953, −12.28154252496880551762487242419, −10.71954135442995424517165341355, −9.340089641217331537652804643219, −8.655822782872438977934600217975, −7.55749757830037023328119719754, −6.45407631261374270091162846267, −5.12557757898750984380487762630, −4.21974935107755913754805952240, −2.54191700186214843862740036992,
1.98938160297877168541132358513, 3.19530636526105843873091342773, 4.49786655818848706982819125158, 6.01509269021380749948007439852, 6.95723420174398810834392927695, 8.634304182065239452399367481473, 9.534249232954428753056993028522, 10.61938907643256199797080041673, 11.28794646866006950676033874108, 12.53007939098664441067859948295
