L(s) = 1 | − 0.382i·2-s + 3.15i·3-s + 3.85·4-s + 1.20·6-s + 4.26·7-s − 3.00i·8-s − 0.938·9-s − 9.61·11-s + 12.1i·12-s + 22.7i·13-s − 1.63i·14-s + 14.2·16-s + 25.9·17-s + 0.359i·18-s + (8.03 − 17.2i)19-s + ⋯ |
L(s) = 1 | − 0.191i·2-s + 1.05i·3-s + 0.963·4-s + 0.200·6-s + 0.609·7-s − 0.375i·8-s − 0.104·9-s − 0.874·11-s + 1.01i·12-s + 1.75i·13-s − 0.116i·14-s + 0.891·16-s + 1.52·17-s + 0.0199i·18-s + (0.422 − 0.906i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.422 - 0.906i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.422 - 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.375487959\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.375487959\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 19 | \( 1 + (-8.03 + 17.2i)T \) |
good | 2 | \( 1 + 0.382iT - 4T^{2} \) |
| 3 | \( 1 - 3.15iT - 9T^{2} \) |
| 7 | \( 1 - 4.26T + 49T^{2} \) |
| 11 | \( 1 + 9.61T + 121T^{2} \) |
| 13 | \( 1 - 22.7iT - 169T^{2} \) |
| 17 | \( 1 - 25.9T + 289T^{2} \) |
| 23 | \( 1 + 24.5T + 529T^{2} \) |
| 29 | \( 1 + 21.2iT - 841T^{2} \) |
| 31 | \( 1 + 6.67iT - 961T^{2} \) |
| 37 | \( 1 - 35.0iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 72.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 50.2T + 1.84e3T^{2} \) |
| 47 | \( 1 + 36.1T + 2.20e3T^{2} \) |
| 53 | \( 1 + 2.29iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 19.0iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 50.6T + 3.72e3T^{2} \) |
| 67 | \( 1 + 9.20iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 116. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 49.2T + 5.32e3T^{2} \) |
| 79 | \( 1 - 87.5iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 33.8T + 6.88e3T^{2} \) |
| 89 | \( 1 - 7.05iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 16.0iT - 9.40e3T^{2} \) |
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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
−11.07234628497242249514537604110, −9.947267284023194696978785094614, −9.642478163230072857013811928285, −8.187974868638990272508649844469, −7.37639770123488995142607646995, −6.26045876266139873725272478948, −5.08924034902404393792631227228, −4.17921564253165098499406296401, −2.95987253065211638552246734153, −1.62782633200982917378324098662,
1.05327680450258020517164190332, 2.22374595699555005591586123468, 3.40539771734937494842003930549, 5.44107317519887592165529116215, 5.85023154232092599645446004057, 7.27922537658740743868046141593, 7.74443188053334782183433088774, 8.236397730708522616369068667949, 10.15043296539432769254207005689, 10.51897465374985878477170792827