| L(s) = 1 | + (−0.0822 + 0.0448i)3-s + (−1.59 − 1.57i)5-s + (3.12 + 4.17i)7-s + (−1.61 + 2.51i)9-s + (−3.36 + 1.53i)11-s + (−1.74 − 1.31i)13-s + (0.201 + 0.0578i)15-s + (−0.442 − 6.18i)17-s + (−2.05 − 2.37i)19-s + (−0.443 − 0.202i)21-s + (−4.77 + 0.466i)23-s + (0.0590 + 4.99i)25-s + (0.0400 − 0.559i)27-s + (−1.75 − 1.52i)29-s + (−0.693 + 0.203i)31-s + ⋯ |
| L(s) = 1 | + (−0.0474 + 0.0259i)3-s + (−0.711 − 0.702i)5-s + (1.18 + 1.57i)7-s + (−0.539 + 0.838i)9-s + (−1.01 + 0.462i)11-s + (−0.485 − 0.363i)13-s + (0.0519 + 0.0149i)15-s + (−0.107 − 1.49i)17-s + (−0.471 − 0.544i)19-s + (−0.0968 − 0.0442i)21-s + (−0.995 + 0.0973i)23-s + (0.0118 + 0.999i)25-s + (0.00770 − 0.107i)27-s + (−0.326 − 0.283i)29-s + (−0.124 + 0.0365i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.937 - 0.348i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.937 - 0.348i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0710132 + 0.394832i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0710132 + 0.394832i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (1.59 + 1.57i)T \) |
| 23 | \( 1 + (4.77 - 0.466i)T \) |
| good | 3 | \( 1 + (0.0822 - 0.0448i)T + (1.62 - 2.52i)T^{2} \) |
| 7 | \( 1 + (-3.12 - 4.17i)T + (-1.97 + 6.71i)T^{2} \) |
| 11 | \( 1 + (3.36 - 1.53i)T + (7.20 - 8.31i)T^{2} \) |
| 13 | \( 1 + (1.74 + 1.31i)T + (3.66 + 12.4i)T^{2} \) |
| 17 | \( 1 + (0.442 + 6.18i)T + (-16.8 + 2.41i)T^{2} \) |
| 19 | \( 1 + (2.05 + 2.37i)T + (-2.70 + 18.8i)T^{2} \) |
| 29 | \( 1 + (1.75 + 1.52i)T + (4.12 + 28.7i)T^{2} \) |
| 31 | \( 1 + (0.693 - 0.203i)T + (26.0 - 16.7i)T^{2} \) |
| 37 | \( 1 + (8.16 - 1.77i)T + (33.6 - 15.3i)T^{2} \) |
| 41 | \( 1 + (4.34 - 2.79i)T + (17.0 - 37.2i)T^{2} \) |
| 43 | \( 1 + (-0.396 - 0.726i)T + (-23.2 + 36.1i)T^{2} \) |
| 47 | \( 1 + (-6.39 - 6.39i)T + 47iT^{2} \) |
| 53 | \( 1 + (8.20 - 6.14i)T + (14.9 - 50.8i)T^{2} \) |
| 59 | \( 1 + (1.11 - 0.160i)T + (56.6 - 16.6i)T^{2} \) |
| 61 | \( 1 + (-1.41 - 4.82i)T + (-51.3 + 32.9i)T^{2} \) |
| 67 | \( 1 + (-1.29 - 3.47i)T + (-50.6 + 43.8i)T^{2} \) |
| 71 | \( 1 + (-5.27 + 11.5i)T + (-46.4 - 53.6i)T^{2} \) |
| 73 | \( 1 + (-1.25 - 0.0898i)T + (72.2 + 10.3i)T^{2} \) |
| 79 | \( 1 + (-0.267 - 1.85i)T + (-75.7 + 22.2i)T^{2} \) |
| 83 | \( 1 + (3.06 + 14.0i)T + (-75.4 + 34.4i)T^{2} \) |
| 89 | \( 1 + (-11.9 - 3.51i)T + (74.8 + 48.1i)T^{2} \) |
| 97 | \( 1 + (2.27 - 10.4i)T + (-88.2 - 40.2i)T^{2} \) |
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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
−10.61761768862186870185087348233, −9.390363081212865519760862598470, −8.672751908697642576516432988986, −7.951768558416550349096230939954, −7.48719404105871459278791962283, −5.78752238205292435134292847445, −4.95488932412350284795025746043, −4.77842437904065496442926710895, −2.82368650160340843841092887825, −2.02811439216000820647435288693,
0.17771206971871465201618942020, 1.93278752926885392969164399778, 3.59277292935272268519315278889, 4.02997397218740028422058592695, 5.26429979576127955205713367309, 6.44057390786567298856195738343, 7.23678177773250658883180649469, 8.076789350369246002960252642083, 8.478071815680230749354709961865, 10.09871057582035465861633549091
