L(s) = 1 | + 2.36i·2-s + (7.81 + 4.46i)3-s + 10.4·4-s − 9.92i·5-s + (−10.5 + 18.4i)6-s − 13.3·7-s + 62.3i·8-s + (41.1 + 69.7i)9-s + 23.4·10-s − 36.4i·11-s + (81.4 + 46.5i)12-s − 138.·13-s − 31.4i·14-s + (44.3 − 77.5i)15-s + 19.5·16-s − 445. i·17-s + ⋯ |
L(s) = 1 | + 0.590i·2-s + (0.868 + 0.496i)3-s + 0.651·4-s − 0.397i·5-s + (−0.292 + 0.512i)6-s − 0.272·7-s + 0.974i·8-s + (0.507 + 0.861i)9-s + 0.234·10-s − 0.301i·11-s + (0.565 + 0.323i)12-s − 0.820·13-s − 0.160i·14-s + (0.197 − 0.344i)15-s + 0.0765·16-s − 1.54i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.496 - 0.868i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.496 - 0.868i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(1.71629 + 0.996073i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.71629 + 0.996073i\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-7.81 - 4.46i)T \) |
| 11 | \( 1 + 36.4iT \) |
good | 2 | \( 1 - 2.36iT - 16T^{2} \) |
| 5 | \( 1 + 9.92iT - 625T^{2} \) |
| 7 | \( 1 + 13.3T + 2.40e3T^{2} \) |
| 13 | \( 1 + 138.T + 2.85e4T^{2} \) |
| 17 | \( 1 + 445. iT - 8.35e4T^{2} \) |
| 19 | \( 1 + 151.T + 1.30e5T^{2} \) |
| 23 | \( 1 + 661. iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 130. iT - 7.07e5T^{2} \) |
| 31 | \( 1 + 1.03e3T + 9.23e5T^{2} \) |
| 37 | \( 1 - 1.11e3T + 1.87e6T^{2} \) |
| 41 | \( 1 - 1.27e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 + 1.13e3T + 3.41e6T^{2} \) |
| 47 | \( 1 - 915. iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 1.30e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 - 1.57e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 + 5.83e3T + 1.38e7T^{2} \) |
| 67 | \( 1 - 7.20e3T + 2.01e7T^{2} \) |
| 71 | \( 1 - 9.59e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 1.05e4T + 2.83e7T^{2} \) |
| 79 | \( 1 - 1.91e3T + 3.89e7T^{2} \) |
| 83 | \( 1 - 1.08e4iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 6.02e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 199.T + 8.85e7T^{2} \) |
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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
−16.21900284343242111935980012761, −15.00110970568927874534487315728, −14.16914254838973826319584341901, −12.65539264324210000192836442269, −11.04907554005211469054199086061, −9.558063129055871438770092463811, −8.248272537107441248074513775262, −6.93824499462606343664870455938, −4.95715528684772585448685479544, −2.69006146467603872202887313513,
1.95668260461259627631196430166, 3.49339954739542190199337775099, 6.52889092808864993222027639677, 7.71135506207232597796684314975, 9.492405049042558064845578939140, 10.71885008096959976289958021110, 12.23674065498288657578000022908, 13.06592583682663760712407863797, 14.66390370991988198037180473808, 15.39860919471426603540463692830