L(s) = 1 | + (−0.5 − 0.866i)3-s − i·5-s + (0.866 + 0.5i)7-s + (−0.5 + 0.866i)9-s + (0.866 − 0.5i)11-s + (0.866 − 0.5i)15-s + (0.5 − 0.866i)17-s + (0.866 + 0.5i)19-s − i·21-s + (−0.5 − 0.866i)23-s − 25-s + 27-s + (0.5 + 0.866i)29-s − i·31-s + (−0.866 − 0.5i)33-s + ⋯ |
L(s) = 1 | + (−0.5 − 0.866i)3-s − i·5-s + (0.866 + 0.5i)7-s + (−0.5 + 0.866i)9-s + (0.866 − 0.5i)11-s + (0.866 − 0.5i)15-s + (0.5 − 0.866i)17-s + (0.866 + 0.5i)19-s − i·21-s + (−0.5 − 0.866i)23-s − 25-s + 27-s + (0.5 + 0.866i)29-s − i·31-s + (−0.866 − 0.5i)33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 104 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0386i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 104 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0386i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9676930934 + 0.01868545226i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9676930934 + 0.01868545226i\) |
\(L(1)\) |
\(\approx\) |
\(0.9895344955 - 0.04741307888i\) |
\(L(1)\) |
\(\approx\) |
\(0.9895344955 - 0.04741307888i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 13 | \( 1 \) |
good | 3 | \( 1 + (-0.5 - 0.866i)T \) |
| 5 | \( 1 - iT \) |
| 7 | \( 1 + (0.866 + 0.5i)T \) |
| 11 | \( 1 + (0.866 - 0.5i)T \) |
| 17 | \( 1 + (0.5 - 0.866i)T \) |
| 19 | \( 1 + (0.866 + 0.5i)T \) |
| 23 | \( 1 + (-0.5 - 0.866i)T \) |
| 29 | \( 1 + (0.5 + 0.866i)T \) |
| 31 | \( 1 - iT \) |
| 37 | \( 1 + (-0.866 + 0.5i)T \) |
| 41 | \( 1 + (-0.866 + 0.5i)T \) |
| 43 | \( 1 + (0.5 - 0.866i)T \) |
| 47 | \( 1 - iT \) |
| 53 | \( 1 - T \) |
| 59 | \( 1 + (-0.866 - 0.5i)T \) |
| 61 | \( 1 + (0.5 - 0.866i)T \) |
| 67 | \( 1 + (-0.866 + 0.5i)T \) |
| 71 | \( 1 + (-0.866 - 0.5i)T \) |
| 73 | \( 1 + iT \) |
| 79 | \( 1 - T \) |
| 83 | \( 1 - iT \) |
| 89 | \( 1 + (0.866 - 0.5i)T \) |
| 97 | \( 1 + (0.866 + 0.5i)T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−29.67088626968080384642172054533, −28.36824073027195341314835986164, −27.82994708388201683550815094466, −26.97229120617451561385491599012, −25.75290764696992086175780384729, −24.404997776764095848428965546513, −23.616043850635802318867785615686, −22.44645362370606304567226385404, −21.304634779364547368921749817581, −20.560583080944362885011377532157, −19.6258184645378278088432143141, −17.585002376213150656821001283301, −17.20752264594623899232433746066, −16.0652198004158634182468935857, −14.97326066555791546669041233951, −13.78638108134130338567754359446, −12.20304890706396478193799566499, −11.40637852192964635028822365527, −10.03328497028333113217357198847, −9.057700278338838841455724834992, −7.726325132990709017264139754199, −5.91217399796355305063261290517, −4.745789265159576074864739481749, −3.88289611297301237520303759556, −1.34652327488651930984610786917,
1.59032611547197458758741245377, 3.09990239646157923027105161055, 5.15410362658805478349830946732, 6.35929671034005700082219982111, 7.38745362084218012781074582067, 8.59519333488089384551014010966, 10.383885290763299354566749952661, 11.55236638727003472255788039549, 12.13309307321626392543647890515, 13.9680920582600622766394562588, 14.40403276133497482632432403486, 16.043929047276202641003336346385, 17.34950891602712817082902158690, 18.33007020215300278533057198980, 18.849736384997758996442238657410, 20.2157200645770241377150661447, 21.7749373869791103373462180086, 22.486195200965964923592838740276, 23.55508939209076775713009567348, 24.65588424733617818771483893448, 25.32766843884309266113030475202, 26.858448672027796722145402412488, 27.653779854583120858001059371985, 28.927452391209617463888155106649, 29.86577340150726344981344796418