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
−11.44940002748752511373316145660, −10.45194347493837947526748928188, −9.749769005519523823045164013631, −8.306349409840086404648726004846, −7.42353862017663581087153269025, −6.64932358779491242492028023100, −5.66143694907882928659705760391, −4.48586366069119514901987849258, −3.36660412282326578647472518836, −2.22825371439739504892964493047,
1.05472343699070841405366800854, 2.86101686044697262487783981226, 4.05734633762596939404687052940, 5.09703690885144532481940228761, 5.84984733229303615030047353137, 7.15446788884031181231453552770, 8.182253720530650720837679442372, 9.060006942255340361455474293629, 10.11437281746297494644901027210, 11.04145198055370416025784515829