L(s) = 1 | + (0.951 − 0.309i)2-s + (0.809 − 0.587i)4-s + (−0.420 − 2.19i)5-s − 0.407i·7-s + (0.587 − 0.809i)8-s + (−1.07 − 1.95i)10-s + (−0.930 − 2.86i)11-s + (−0.172 − 0.0560i)13-s + (−0.125 − 0.387i)14-s + (0.309 − 0.951i)16-s + (1.82 − 2.50i)17-s + (1.15 + 0.840i)19-s + (−1.63 − 1.52i)20-s + (−1.76 − 2.43i)22-s + (1.06 − 0.345i)23-s + ⋯ |
L(s) = 1 | + (0.672 − 0.218i)2-s + (0.404 − 0.293i)4-s + (−0.187 − 0.982i)5-s − 0.153i·7-s + (0.207 − 0.286i)8-s + (−0.340 − 0.619i)10-s + (−0.280 − 0.863i)11-s + (−0.0478 − 0.0155i)13-s + (−0.0336 − 0.103i)14-s + (0.0772 − 0.237i)16-s + (0.441 − 0.607i)17-s + (0.265 + 0.192i)19-s + (−0.364 − 0.342i)20-s + (−0.377 − 0.519i)22-s + (0.221 − 0.0719i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.126 + 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.126 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.44724 - 1.27462i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.44724 - 1.27462i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.951 + 0.309i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (0.420 + 2.19i)T \) |
good | 7 | \( 1 + 0.407iT - 7T^{2} \) |
| 11 | \( 1 + (0.930 + 2.86i)T + (-8.89 + 6.46i)T^{2} \) |
| 13 | \( 1 + (0.172 + 0.0560i)T + (10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (-1.82 + 2.50i)T + (-5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (-1.15 - 0.840i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (-1.06 + 0.345i)T + (18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (0.127 - 0.0928i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (-6.96 - 5.06i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (4.19 + 1.36i)T + (29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (-2.54 + 7.83i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 - 10.6iT - 43T^{2} \) |
| 47 | \( 1 + (1.62 + 2.23i)T + (-14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (-4.93 - 6.78i)T + (-16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (2.21 - 6.82i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (-2.09 - 6.44i)T + (-49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + (-2.32 + 3.19i)T + (-20.7 - 63.7i)T^{2} \) |
| 71 | \( 1 + (-2.49 + 1.81i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (8.18 - 2.66i)T + (59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (-5.59 + 4.06i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (8.70 - 11.9i)T + (-25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + (-4.52 - 13.9i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (9.59 + 13.2i)T + (-29.9 + 92.2i)T^{2} \) |
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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
−11.04145198055370416025784515829, −10.11437281746297494644901027210, −9.060006942255340361455474293629, −8.182253720530650720837679442372, −7.15446788884031181231453552770, −5.84984733229303615030047353137, −5.09703690885144532481940228761, −4.05734633762596939404687052940, −2.86101686044697262487783981226, −1.05472343699070841405366800854,
2.22825371439739504892964493047, 3.36660412282326578647472518836, 4.48586366069119514901987849258, 5.66143694907882928659705760391, 6.64932358779491242492028023100, 7.42353862017663581087153269025, 8.306349409840086404648726004846, 9.749769005519523823045164013631, 10.45194347493837947526748928188, 11.44940002748752511373316145660