L(s) = 1 | + 1.60i·2-s + 1.41·4-s − 6.21i·5-s + 11.1·7-s + 8.70i·8-s + 9.98·10-s + 3.31i·11-s − 17.7·13-s + 17.9i·14-s − 8.31·16-s + 8.53i·17-s + 16.4·19-s − 8.80i·20-s − 5.32·22-s − 25.0i·23-s + ⋯ |
L(s) = 1 | + 0.803i·2-s + 0.354·4-s − 1.24i·5-s + 1.59·7-s + 1.08i·8-s + 0.998·10-s + 0.301i·11-s − 1.36·13-s + 1.27i·14-s − 0.519·16-s + 0.502i·17-s + 0.866·19-s − 0.440i·20-s − 0.242·22-s − 1.09i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.55885 + 0.495461i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.55885 + 0.495461i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 - 3.31iT \) |
good | 2 | \( 1 - 1.60iT - 4T^{2} \) |
| 5 | \( 1 + 6.21iT - 25T^{2} \) |
| 7 | \( 1 - 11.1T + 49T^{2} \) |
| 13 | \( 1 + 17.7T + 169T^{2} \) |
| 17 | \( 1 - 8.53iT - 289T^{2} \) |
| 19 | \( 1 - 16.4T + 361T^{2} \) |
| 23 | \( 1 + 25.0iT - 529T^{2} \) |
| 29 | \( 1 - 9.46iT - 841T^{2} \) |
| 31 | \( 1 + 46.1T + 961T^{2} \) |
| 37 | \( 1 + 37.4T + 1.36e3T^{2} \) |
| 41 | \( 1 + 51.8iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 55.6T + 1.84e3T^{2} \) |
| 47 | \( 1 - 37.8iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 58.5iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 44.6iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 10.1T + 3.72e3T^{2} \) |
| 67 | \( 1 + 58.3T + 4.48e3T^{2} \) |
| 71 | \( 1 - 75.1iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 112.T + 5.32e3T^{2} \) |
| 79 | \( 1 - 52.2T + 6.24e3T^{2} \) |
| 83 | \( 1 - 18.0iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 136. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 127.T + 9.40e3T^{2} \) |
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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
−14.18294234559468313230571516208, −12.55170884197239881904114022224, −11.82797613058110814962638024829, −10.63866475758284980246372358087, −8.994919424395936775496395071358, −8.066783650643853684855015883076, −7.20282649013378807408609092704, −5.37375796084896131956724437492, −4.76445427698419078254017929745, −1.85412047241642429588746034994,
1.95146832998413090276851987671, 3.30269668537560490287510187510, 5.16861813201072160803308590498, 6.98113036203136479296606225408, 7.73487934914287659853159935857, 9.607381633568827573141799499822, 10.63865487431450717929756649472, 11.43874572246185817232559991725, 11.97023002149477928839766795631, 13.61430173996342180849064753950