L(s) = 1 | + (0.826 + 0.563i)5-s + (0.988 − 0.149i)11-s + (−0.988 + 0.149i)13-s + (−0.733 + 0.680i)17-s + (0.5 − 0.866i)19-s + (−0.955 + 0.294i)23-s + (0.365 + 0.930i)25-s + (−0.733 + 0.680i)29-s − 31-s + (0.955 + 0.294i)37-s + (0.0747 − 0.997i)41-s + (−0.0747 − 0.997i)43-s + (−0.623 − 0.781i)47-s + (0.955 − 0.294i)53-s + (0.900 + 0.433i)55-s + ⋯ |
L(s) = 1 | + (0.826 + 0.563i)5-s + (0.988 − 0.149i)11-s + (−0.988 + 0.149i)13-s + (−0.733 + 0.680i)17-s + (0.5 − 0.866i)19-s + (−0.955 + 0.294i)23-s + (0.365 + 0.930i)25-s + (−0.733 + 0.680i)29-s − 31-s + (0.955 + 0.294i)37-s + (0.0747 − 0.997i)41-s + (−0.0747 − 0.997i)43-s + (−0.623 − 0.781i)47-s + (0.955 − 0.294i)53-s + (0.900 + 0.433i)55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.110 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.110 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8245807520 - 0.9210542429i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8245807520 - 0.9210542429i\) |
\(L(1)\) |
\(\approx\) |
\(1.082581346 + 0.03106472715i\) |
\(L(1)\) |
\(\approx\) |
\(1.082581346 + 0.03106472715i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (0.826 + 0.563i)T \) |
| 11 | \( 1 + (0.988 - 0.149i)T \) |
| 13 | \( 1 + (-0.988 + 0.149i)T \) |
| 17 | \( 1 + (-0.733 + 0.680i)T \) |
| 19 | \( 1 + (0.5 - 0.866i)T \) |
| 23 | \( 1 + (-0.955 + 0.294i)T \) |
| 29 | \( 1 + (-0.733 + 0.680i)T \) |
| 31 | \( 1 - T \) |
| 37 | \( 1 + (0.955 + 0.294i)T \) |
| 41 | \( 1 + (0.0747 - 0.997i)T \) |
| 43 | \( 1 + (-0.0747 - 0.997i)T \) |
| 47 | \( 1 + (-0.623 - 0.781i)T \) |
| 53 | \( 1 + (0.955 - 0.294i)T \) |
| 59 | \( 1 + (0.900 + 0.433i)T \) |
| 61 | \( 1 + (-0.222 - 0.974i)T \) |
| 67 | \( 1 - T \) |
| 71 | \( 1 + (0.222 - 0.974i)T \) |
| 73 | \( 1 + (-0.988 - 0.149i)T \) |
| 79 | \( 1 - T \) |
| 83 | \( 1 + (0.988 + 0.149i)T \) |
| 89 | \( 1 + (0.365 + 0.930i)T \) |
| 97 | \( 1 + (-0.5 - 0.866i)T \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−20.06834469236347683508259084633, −19.87464855776225910948505189963, −18.672274037313442363513682125420, −17.90817219668800610160519606504, −17.38368958063050102604409277263, −16.483961621691300691474441337046, −16.161747362680878460582160946402, −14.69350068357662980059202239484, −14.52695615028932648864563244563, −13.48342256195995601534024058984, −12.87143622568947573687813794277, −12.0084232386431138138197604823, −11.43196462606792920258682248556, −10.22469305203516608263358565019, −9.58185962927631848789389747096, −9.132061993030065831609262360907, −8.065793957729111423576276080664, −7.2550511146255282711029191362, −6.27216810741187350004080192377, −5.6527944550004907885085385562, −4.68075058687163186280878092997, −4.01229252715079911928121204082, −2.71289950233348411086417244311, −1.91799494657134245878955236006, −1.01020289290334784822875733234,
0.21685068243491896309279339776, 1.650564676077797808188897266554, 2.233338480623196146441090618267, 3.32091375299577650632827555771, 4.17524233630884787024284204988, 5.25373077345672779198384883374, 5.97638476009744567706736587618, 6.88273970068495300746687292858, 7.336229602919190612604901182420, 8.6573590867778467219786195204, 9.31459953840648705475195838455, 9.96588826969819379187316897261, 10.83559401384872218069360269611, 11.54182898346685861362704873159, 12.37698012519522241132963774223, 13.31102238831413405746225385401, 13.89816032297203620457745967269, 14.74331833578006501214279637738, 15.129028617787050990815520835739, 16.35335820532050795085998642872, 16.9882227618593090409920882697, 17.711696239857111654138388042555, 18.20726004526695172280298125546, 19.23177336150620755632375337052, 19.80464520792068704182108885652