L(s) = 1 | + (−0.5 + 0.866i)2-s + (0.500 + 0.866i)4-s − 3·8-s + (1.5 − 2.59i)9-s + (−2 − 3.46i)11-s + (0.500 − 0.866i)16-s + (1.5 + 2.59i)18-s + 3.99·22-s + (−4 + 6.92i)23-s + (2.5 + 4.33i)25-s + 2·29-s + (−2.50 − 4.33i)32-s + 3·36-s + (3 − 5.19i)37-s − 12·43-s + (1.99 − 3.46i)44-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (0.250 + 0.433i)4-s − 1.06·8-s + (0.5 − 0.866i)9-s + (−0.603 − 1.04i)11-s + (0.125 − 0.216i)16-s + (0.353 + 0.612i)18-s + 0.852·22-s + (−0.834 + 1.44i)23-s + (0.5 + 0.866i)25-s + 0.371·29-s + (−0.441 − 0.765i)32-s + 0.5·36-s + (0.493 − 0.854i)37-s − 1.82·43-s + (0.301 − 0.522i)44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.654693 + 0.324549i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.654693 + 0.324549i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
good | 2 | \( 1 + (0.5 - 0.866i)T + (-1 - 1.73i)T^{2} \) |
| 3 | \( 1 + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (2 + 3.46i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (4 - 6.92i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-3 + 5.19i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 12T + 43T^{2} \) |
| 47 | \( 1 + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-5 - 8.66i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (2 + 3.46i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 16T + 71T^{2} \) |
| 73 | \( 1 + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (4 - 6.92i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 97T^{2} \) |
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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
−15.85212576093522570832154229519, −15.08039720005677721664988852131, −13.54999095046078507734294954532, −12.35110043118596423755352575705, −11.22716252147463454174169575444, −9.576821537056004788205838341040, −8.359819327199796891744287885900, −7.16251823162741675847358746285, −5.83768404749240717606545266475, −3.42987312905157534010236914006,
2.26623532713692530088546710173, 4.85098688399215011214238702043, 6.65752375968195805720507732929, 8.255622698155190429547567793677, 9.963629133540794207882785141408, 10.49157349351209535688193050598, 11.89293230051654397708049852819, 12.97664701560853149160404088547, 14.43344061342251990602688111731, 15.49241818306105495547348293749