L(s) = 1 | + (−0.281 − 0.959i)2-s + i·3-s + (−0.841 + 0.540i)4-s + (0.959 − 0.281i)6-s + (−0.909 + 0.415i)7-s + (0.755 + 0.654i)8-s − 9-s + (−0.540 − 0.841i)12-s + (−0.540 − 0.841i)13-s + (0.654 + 0.755i)14-s + (0.415 − 0.909i)16-s + (−0.989 − 0.142i)17-s + (0.281 + 0.959i)18-s + (−0.142 − 0.989i)19-s + (−0.415 − 0.909i)21-s + ⋯ |
L(s) = 1 | + (−0.281 − 0.959i)2-s + i·3-s + (−0.841 + 0.540i)4-s + (0.959 − 0.281i)6-s + (−0.909 + 0.415i)7-s + (0.755 + 0.654i)8-s − 9-s + (−0.540 − 0.841i)12-s + (−0.540 − 0.841i)13-s + (0.654 + 0.755i)14-s + (0.415 − 0.909i)16-s + (−0.989 − 0.142i)17-s + (0.281 + 0.959i)18-s + (−0.142 − 0.989i)19-s + (−0.415 − 0.909i)21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.347 - 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.347 - 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5531908684 - 0.3848454828i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5531908684 - 0.3848454828i\) |
\(L(1)\) |
\(\approx\) |
\(0.6771423301 - 0.1331872604i\) |
\(L(1)\) |
\(\approx\) |
\(0.6771423301 - 0.1331872604i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-0.281 - 0.959i)T \) |
| 3 | \( 1 + iT \) |
| 7 | \( 1 + (-0.909 + 0.415i)T \) |
| 13 | \( 1 + (-0.540 - 0.841i)T \) |
| 17 | \( 1 + (-0.989 - 0.142i)T \) |
| 19 | \( 1 + (-0.142 - 0.989i)T \) |
| 23 | \( 1 + (0.909 + 0.415i)T \) |
| 29 | \( 1 + (-0.142 - 0.989i)T \) |
| 31 | \( 1 + (0.841 + 0.540i)T \) |
| 37 | \( 1 + (0.540 - 0.841i)T \) |
| 41 | \( 1 + (0.959 - 0.281i)T \) |
| 43 | \( 1 + (0.755 + 0.654i)T \) |
| 47 | \( 1 + (-0.281 + 0.959i)T \) |
| 53 | \( 1 + (-0.909 + 0.415i)T \) |
| 59 | \( 1 + (0.959 + 0.281i)T \) |
| 61 | \( 1 + (0.959 + 0.281i)T \) |
| 67 | \( 1 + (0.281 + 0.959i)T \) |
| 71 | \( 1 + (-0.142 - 0.989i)T \) |
| 73 | \( 1 + (-0.909 - 0.415i)T \) |
| 79 | \( 1 + (-0.654 - 0.755i)T \) |
| 83 | \( 1 + (0.909 - 0.415i)T \) |
| 89 | \( 1 + (0.142 - 0.989i)T \) |
| 97 | \( 1 + (0.755 + 0.654i)T \) |
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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
−23.426798401527668814947533701480, −22.69358683759152857603060250086, −22.021680972305454362155802223585, −20.49498798414732849058928126018, −19.49518228784051248734679627188, −19.02551474273577214661737170780, −18.27516642050439528072022671875, −17.22079879201376620629019146168, −16.77617401503851576755095350902, −15.8780158137602574898010734619, −14.72868599185053349580208463281, −14.07107941811418007292106774266, −13.14694532750874451176540754461, −12.63343556434730763714050956595, −11.35709592101510554340821032530, −10.20271046219430740679450175143, −9.24349767044627907408504243331, −8.44193880068536104663551997836, −7.416411283787925630524481130018, −6.70328025966131499477366786036, −6.18438174997719670348907058020, −4.94293502010080836490009220577, −3.77391320195253563299716652807, −2.32346365027550042673339473619, −0.96601303931366632349774806717,
0.476831618486539139862741587344, 2.5412085586115087325738546413, 2.96510851916012533022667189314, 4.16403107658710618252198055085, 4.967518004016579471219295289002, 6.06073434918627490036719098799, 7.51459358368439193397455540167, 8.76562952217739330096236097383, 9.30726374237163433695288544352, 10.0530359753590199771768551366, 10.92118602236425629902833778364, 11.626132086616590392249191177719, 12.75050020489787689514421831155, 13.35330987339987454914396804642, 14.55591961181377738821721073505, 15.51963396855265759930374959000, 16.167422505941842153126010849232, 17.426960121601812987027850301784, 17.6903216977644322126918310740, 19.27873775570723357218646780253, 19.512693630047384395361749476238, 20.478180553132190513260714058955, 21.24963447031523273517578500127, 22.059668671584115883838524247521, 22.531683010548264006521487347552