| L(s) = 1 | + (0.962 − 1.75i)2-s + (−2.14 − 3.37i)4-s − 1.68·5-s + (5.30 + 4.56i)7-s + (−7.98 + 0.513i)8-s + (−1.61 + 2.94i)10-s − 11.9i·11-s − 25.3·13-s + (13.1 − 4.91i)14-s + (−6.78 + 14.4i)16-s − 11.7i·17-s − 0.0390·19-s + (3.61 + 5.67i)20-s + (−21.0 − 11.5i)22-s − 26.6·23-s + ⋯ |
| L(s) = 1 | + (0.481 − 0.876i)2-s + (−0.536 − 0.843i)4-s − 0.336·5-s + (0.758 + 0.651i)7-s + (−0.997 + 0.0641i)8-s + (−0.161 + 0.294i)10-s − 1.09i·11-s − 1.94·13-s + (0.936 − 0.351i)14-s + (−0.424 + 0.905i)16-s − 0.689i·17-s − 0.00205·19-s + (0.180 + 0.283i)20-s + (−0.955 − 0.524i)22-s − 1.15·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.798 - 0.601i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.798 - 0.601i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.6453010317\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6453010317\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.962 + 1.75i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-5.30 - 4.56i)T \) |
| good | 5 | \( 1 + 1.68T + 25T^{2} \) |
| 11 | \( 1 + 11.9iT - 121T^{2} \) |
| 13 | \( 1 + 25.3T + 169T^{2} \) |
| 17 | \( 1 + 11.7iT - 289T^{2} \) |
| 19 | \( 1 + 0.0390T + 361T^{2} \) |
| 23 | \( 1 + 26.6T + 529T^{2} \) |
| 29 | \( 1 + 46.0iT - 841T^{2} \) |
| 31 | \( 1 - 37.9iT - 961T^{2} \) |
| 37 | \( 1 - 19.5iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 41.0iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 23.0iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 56.6iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 56.2iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 64.4T + 3.48e3T^{2} \) |
| 61 | \( 1 - 52.2T + 3.72e3T^{2} \) |
| 67 | \( 1 + 55.1iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 8.38T + 5.04e3T^{2} \) |
| 73 | \( 1 + 117. iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 13.2T + 6.24e3T^{2} \) |
| 83 | \( 1 + 19.4T + 6.88e3T^{2} \) |
| 89 | \( 1 + 36.4iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 163. iT - 9.40e3T^{2} \) |
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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
−10.23743949704508649311042447233, −9.521438878593477110625153887164, −8.487590686609276005051259675261, −7.61425502708314380678966905910, −6.11838446032425916335155144363, −5.19718900905223845674300060800, −4.38308015849817942151377508224, −3.02036300006158309570755848756, −2.02965328620293404968344091679, −0.20466646518853949333310692449,
2.20072284574074135959354743404, 3.93113022499019748090954613369, 4.61266716247275766657987793777, 5.53068354970623796980196701732, 6.91350773257823929840889505778, 7.54445978676735571690676268583, 8.118556912933072051819251749476, 9.449746945777853230603902642404, 10.18765138805881131895907516696, 11.47062800001871984024468705757
