L(s) = 1 | + (−0.0747 + 0.997i)2-s + (−0.988 − 0.149i)4-s + (0.222 − 0.974i)5-s + (0.222 − 0.974i)8-s + (0.955 + 0.294i)10-s + (0.900 − 0.433i)11-s + (0.0747 − 0.997i)13-s + (0.955 + 0.294i)16-s + (0.988 − 0.149i)17-s + (−0.5 − 0.866i)19-s + (−0.365 + 0.930i)20-s + (0.365 + 0.930i)22-s + (−0.623 + 0.781i)23-s + (−0.900 − 0.433i)25-s + (0.988 + 0.149i)26-s + ⋯ |
L(s) = 1 | + (−0.0747 + 0.997i)2-s + (−0.988 − 0.149i)4-s + (0.222 − 0.974i)5-s + (0.222 − 0.974i)8-s + (0.955 + 0.294i)10-s + (0.900 − 0.433i)11-s + (0.0747 − 0.997i)13-s + (0.955 + 0.294i)16-s + (0.988 − 0.149i)17-s + (−0.5 − 0.866i)19-s + (−0.365 + 0.930i)20-s + (0.365 + 0.930i)22-s + (−0.623 + 0.781i)23-s + (−0.900 − 0.433i)25-s + (0.988 + 0.149i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0924 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0924 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9041407249 - 0.8240597576i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9041407249 - 0.8240597576i\) |
\(L(1)\) |
\(\approx\) |
\(0.9402475654 + 0.05932401980i\) |
\(L(1)\) |
\(\approx\) |
\(0.9402475654 + 0.05932401980i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (-0.0747 + 0.997i)T \) |
| 5 | \( 1 + (0.222 - 0.974i)T \) |
| 11 | \( 1 + (0.900 - 0.433i)T \) |
| 13 | \( 1 + (0.0747 - 0.997i)T \) |
| 17 | \( 1 + (0.988 - 0.149i)T \) |
| 19 | \( 1 + (-0.5 - 0.866i)T \) |
| 23 | \( 1 + (-0.623 + 0.781i)T \) |
| 29 | \( 1 + (-0.365 + 0.930i)T \) |
| 31 | \( 1 + (-0.5 - 0.866i)T \) |
| 37 | \( 1 + (0.365 - 0.930i)T \) |
| 41 | \( 1 + (0.733 + 0.680i)T \) |
| 43 | \( 1 + (-0.733 + 0.680i)T \) |
| 47 | \( 1 + (-0.0747 + 0.997i)T \) |
| 53 | \( 1 + (-0.365 - 0.930i)T \) |
| 59 | \( 1 + (0.733 - 0.680i)T \) |
| 61 | \( 1 + (-0.988 + 0.149i)T \) |
| 67 | \( 1 + (-0.5 - 0.866i)T \) |
| 71 | \( 1 + (-0.623 + 0.781i)T \) |
| 73 | \( 1 + (0.826 - 0.563i)T \) |
| 79 | \( 1 + (-0.5 + 0.866i)T \) |
| 83 | \( 1 + (-0.0747 - 0.997i)T \) |
| 89 | \( 1 + (-0.0747 - 0.997i)T \) |
| 97 | \( 1 + (-0.5 - 0.866i)T \) |
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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
−23.69488785610262311219711361846, −22.94653951402684885982040032282, −22.21621699516735433801896662654, −21.48159690587335743017760147802, −20.70230759103889414703550612095, −19.65364991208581518847243688987, −18.86763518242597343857563138699, −18.36888242223027725198944370859, −17.270165879938586227845721504, −16.559694126823541583208304263982, −14.88621753017455379174455558624, −14.344400756043114662577699257554, −13.5843715193238620885853348156, −12.24011481177913498962386491438, −11.77412443975813842013663212737, −10.64236815815089900597194070238, −9.98660013647708322760540160138, −9.11194553323068182203207801076, −7.965232410973739128159188390999, −6.77959541291206151849594622445, −5.76016948575842740785182571853, −4.271227315389967519524885712335, −3.57365640014800014556947893643, −2.31523900960791021905924274112, −1.45342295362495438847734874221,
0.35580085411054145286765578949, 1.39798320963257561210676178530, 3.39907296943967779839377995514, 4.479658985654528054182601527495, 5.50004246327120542217404892036, 6.139098978338522469682820723088, 7.47010879831329853542806608752, 8.26349404227016890224324417454, 9.19004220031118747287402528425, 9.8296896850681429939635869418, 11.2446106617354642402516023645, 12.546221690167154757883770101553, 13.144707380327714234437349373675, 14.1406774526489508822586138765, 14.95732019190054288741930798842, 16.01743258105365513667695872696, 16.62569903621477362545149018535, 17.430091971371441241350587250905, 18.1343601702591645113454329308, 19.369148188254027630972122844732, 20.04515290327985001334324791298, 21.279953185263565619946535798147, 22.02942602824605415255195977820, 22.98688528867698995003450384071, 23.89113955224504643336104546175