L(s) = 1 | + 3.09i·2-s + (1.87 − 2.34i)3-s − 5.59·4-s − 8.72i·5-s + (7.26 + 5.79i)6-s − 11.2·7-s − 4.93i·8-s + (−1.99 − 8.77i)9-s + 27.0·10-s − 9.18i·11-s + (−10.4 + 13.1i)12-s + 7.17·13-s − 34.7i·14-s + (−20.4 − 16.3i)15-s − 7.08·16-s − 10.0i·17-s + ⋯ |
L(s) = 1 | + 1.54i·2-s + (0.623 − 0.781i)3-s − 1.39·4-s − 1.74i·5-s + (1.21 + 0.965i)6-s − 1.60·7-s − 0.617i·8-s + (−0.221 − 0.975i)9-s + 2.70·10-s − 0.834i·11-s + (−0.872 + 1.09i)12-s + 0.551·13-s − 2.48i·14-s + (−1.36 − 1.08i)15-s − 0.442·16-s − 0.590i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.781 + 0.623i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.781 + 0.623i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.20244 - 0.420953i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.20244 - 0.420953i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-1.87 + 2.34i)T \) |
| 59 | \( 1 + 7.68iT \) |
good | 2 | \( 1 - 3.09iT - 4T^{2} \) |
| 5 | \( 1 + 8.72iT - 25T^{2} \) |
| 7 | \( 1 + 11.2T + 49T^{2} \) |
| 11 | \( 1 + 9.18iT - 121T^{2} \) |
| 13 | \( 1 - 7.17T + 169T^{2} \) |
| 17 | \( 1 + 10.0iT - 289T^{2} \) |
| 19 | \( 1 - 27.7T + 361T^{2} \) |
| 23 | \( 1 - 2.46iT - 529T^{2} \) |
| 29 | \( 1 - 42.9iT - 841T^{2} \) |
| 31 | \( 1 - 39.9T + 961T^{2} \) |
| 37 | \( 1 + 42.6T + 1.36e3T^{2} \) |
| 41 | \( 1 - 20.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 56.9T + 1.84e3T^{2} \) |
| 47 | \( 1 + 47.6iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 68.4iT - 2.80e3T^{2} \) |
| 61 | \( 1 + 31.4T + 3.72e3T^{2} \) |
| 67 | \( 1 - 17.2T + 4.48e3T^{2} \) |
| 71 | \( 1 + 42.1iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 45.5T + 5.32e3T^{2} \) |
| 79 | \( 1 - 27.8T + 6.24e3T^{2} \) |
| 83 | \( 1 + 0.101iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 34.2iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 59.1T + 9.40e3T^{2} \) |
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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
−12.79766493523062159632304157500, −11.88403346047008336074738409735, −9.551723110967925492912832234167, −8.943681846543709975885650879814, −8.240797392926697218014307185131, −7.11875359009413283681192511102, −6.13818123739570306186366681126, −5.20415374963244135315048335731, −3.42886288565718987061009368150, −0.72061963044604744534574327416,
2.50447313377578486605655206130, 3.21367901097278985946324725972, 4.03462259862840858005966210893, 6.21772487313621301452087682106, 7.44767764391800677306607560148, 9.235943881263900439593934466265, 10.00293088697666930206580546063, 10.34946241148956147878123534963, 11.32326989486838329183577054427, 12.38953028017564413985828035543