L(s) = 1 | + (−0.788 − 0.661i)2-s + (−0.897 + 1.48i)3-s + (−0.163 − 0.926i)4-s + (0.446 + 1.22i)5-s + (1.68 − 0.573i)6-s + (−0.947 + 1.64i)7-s + (−1.51 + 2.62i)8-s + (−1.38 − 2.65i)9-s + (0.459 − 1.26i)10-s + 5.09i·11-s + (1.51 + 0.589i)12-s + (0.188 − 0.519i)13-s + (1.83 − 0.667i)14-s + (−2.21 − 0.439i)15-s + (1.16 − 0.422i)16-s + (1.79 + 4.93i)17-s + ⋯ |
L(s) = 1 | + (−0.557 − 0.467i)2-s + (−0.518 + 0.855i)3-s + (−0.0816 − 0.463i)4-s + (0.199 + 0.548i)5-s + (0.689 − 0.234i)6-s + (−0.358 + 0.620i)7-s + (−0.535 + 0.926i)8-s + (−0.462 − 0.886i)9-s + (0.145 − 0.399i)10-s + 1.53i·11-s + (0.438 + 0.170i)12-s + (0.0524 − 0.143i)13-s + (0.489 − 0.178i)14-s + (−0.572 − 0.113i)15-s + (0.290 − 0.105i)16-s + (0.435 + 1.19i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0478 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0478 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.410071 + 0.390911i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.410071 + 0.390911i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.897 - 1.48i)T \) |
| 19 | \( 1 + (1.28 - 4.16i)T \) |
good | 2 | \( 1 + (0.788 + 0.661i)T + (0.347 + 1.96i)T^{2} \) |
| 5 | \( 1 + (-0.446 - 1.22i)T + (-3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (0.947 - 1.64i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 - 5.09iT - 11T^{2} \) |
| 13 | \( 1 + (-0.188 + 0.519i)T + (-9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (-1.79 - 4.93i)T + (-13.0 + 10.9i)T^{2} \) |
| 23 | \( 1 + (4.43 - 0.782i)T + (21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (0.744 + 4.22i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + 9.56iT - 31T^{2} \) |
| 37 | \( 1 - 0.822iT - 37T^{2} \) |
| 41 | \( 1 + (4.69 + 3.93i)T + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (0.620 - 3.51i)T + (-40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (-6.74 + 1.18i)T + (44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + (3.87 - 3.24i)T + (9.20 - 52.1i)T^{2} \) |
| 59 | \( 1 + (-1.14 + 6.46i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (-2.20 - 0.802i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (-10.0 - 11.9i)T + (-11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (-0.276 - 0.232i)T + (12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 + (-0.920 + 5.22i)T + (-68.5 - 24.9i)T^{2} \) |
| 79 | \( 1 + (-4.77 - 13.1i)T + (-60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (-6.96 - 4.02i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (0.457 + 2.59i)T + (-83.6 + 30.4i)T^{2} \) |
| 97 | \( 1 + (-1.47 + 1.76i)T + (-16.8 - 95.5i)T^{2} \) |
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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
−12.57444394384049873267902669082, −11.83123267506606903612976569656, −10.67795812463845114854827108825, −10.00037201035597733253376745114, −9.517598909129448263426753367336, −8.172920636619565381165068252555, −6.34555671912650103874906495121, −5.58890798683969788481860365222, −4.09054965423608022739485485805, −2.23087434391682665482962086252,
0.67288756770921952253860439264, 3.23973167000580055434614263466, 5.14268969727741332785340805084, 6.48455519246557871149294797807, 7.25230345721752449214119570037, 8.377036023244943790138302390324, 9.107672703161608683076786655173, 10.57058723985070735451557349067, 11.66113604771792019826209381672, 12.57618392896998451014171650894