L(s) = 1 | + (0.866 + 0.5i)2-s + (−1.36 − 2.36i)3-s + (0.499 + 0.866i)4-s + (1.5 − 0.866i)5-s − 2.73i·6-s + (1 + 1.73i)7-s + 0.999i·8-s + (−2.23 + 3.86i)9-s + 1.73·10-s − 4.73·11-s + (1.36 − 2.36i)12-s + (−3 + 1.73i)13-s + 1.99i·14-s + (−4.09 − 2.36i)15-s + (−0.5 + 0.866i)16-s + (6.69 + 3.86i)17-s + ⋯ |
L(s) = 1 | + (0.612 + 0.353i)2-s + (−0.788 − 1.36i)3-s + (0.249 + 0.433i)4-s + (0.670 − 0.387i)5-s − 1.11i·6-s + (0.377 + 0.654i)7-s + 0.353i·8-s + (−0.744 + 1.28i)9-s + 0.547·10-s − 1.42·11-s + (0.394 − 0.683i)12-s + (−0.832 + 0.480i)13-s + 0.534i·14-s + (−1.05 − 0.610i)15-s + (−0.125 + 0.216i)16-s + (1.62 + 0.937i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.896 + 0.443i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.896 + 0.443i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.04759 - 0.244929i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.04759 - 0.244929i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 - 0.5i)T \) |
| 37 | \( 1 + (5.69 - 2.13i)T \) |
good | 3 | \( 1 + (1.36 + 2.36i)T + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (-1.5 + 0.866i)T + (2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (-1 - 1.73i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + 4.73T + 11T^{2} \) |
| 13 | \( 1 + (3 - 1.73i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-6.69 - 3.86i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.09 + 0.633i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + 4.73iT - 23T^{2} \) |
| 29 | \( 1 + 8.66iT - 29T^{2} \) |
| 31 | \( 1 - 1.26iT - 31T^{2} \) |
| 41 | \( 1 + (4.96 + 8.59i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 - 0.928iT - 43T^{2} \) |
| 47 | \( 1 - 4.73T + 47T^{2} \) |
| 53 | \( 1 + (1.26 - 2.19i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-2.19 - 1.26i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.5 + 0.866i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-5.09 - 8.83i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (1.73 + 3i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 - 4T + 73T^{2} \) |
| 79 | \( 1 + (11.4 - 6.63i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-2.83 + 4.90i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (5.89 + 3.40i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 7.73iT - 97T^{2} \) |
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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
−14.22475780935800700136610272679, −13.23251518755366938539411013694, −12.45141581263461391815434389542, −11.79891918662957083885896220858, −10.22748427003330092383453510609, −8.287451503679754829241604697852, −7.28688928546112566849819576227, −5.85627710721084722349181152484, −5.24088143535986636179282696238, −2.17634053710953356201098501685,
3.21896739138208407882888504935, 4.99630177260622274496114811717, 5.52795291595256257150481002472, 7.48207459630864770676611344149, 9.834822439316199948385952778723, 10.22572874941855798934882086664, 11.13025562576389658608651839975, 12.33829517681194657827370703785, 13.75265949837372006420062638798, 14.62929056130012672685522010831