L(s) = 1 | + (−0.623 − 0.781i)2-s + (−0.222 + 0.974i)4-s + (0.0747 − 0.997i)5-s + (0.900 − 0.433i)8-s + (−0.826 + 0.563i)10-s + (−0.365 + 0.930i)11-s + (−0.365 + 0.930i)13-s + (−0.900 − 0.433i)16-s + (0.955 + 0.294i)17-s + (0.5 + 0.866i)19-s + (0.955 + 0.294i)20-s + (0.955 − 0.294i)22-s + (0.733 + 0.680i)23-s + (−0.988 − 0.149i)25-s + (0.955 − 0.294i)26-s + ⋯ |
L(s) = 1 | + (−0.623 − 0.781i)2-s + (−0.222 + 0.974i)4-s + (0.0747 − 0.997i)5-s + (0.900 − 0.433i)8-s + (−0.826 + 0.563i)10-s + (−0.365 + 0.930i)11-s + (−0.365 + 0.930i)13-s + (−0.900 − 0.433i)16-s + (0.955 + 0.294i)17-s + (0.5 + 0.866i)19-s + (0.955 + 0.294i)20-s + (0.955 − 0.294i)22-s + (0.733 + 0.680i)23-s + (−0.988 − 0.149i)25-s + (0.955 − 0.294i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0142i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0142i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8354211956 + 0.005951468229i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8354211956 + 0.005951468229i\) |
\(L(1)\) |
\(\approx\) |
\(0.7456351916 - 0.1726085396i\) |
\(L(1)\) |
\(\approx\) |
\(0.7456351916 - 0.1726085396i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (-0.623 - 0.781i)T \) |
| 5 | \( 1 + (0.0747 - 0.997i)T \) |
| 11 | \( 1 + (-0.365 + 0.930i)T \) |
| 13 | \( 1 + (-0.365 + 0.930i)T \) |
| 17 | \( 1 + (0.955 + 0.294i)T \) |
| 19 | \( 1 + (0.5 + 0.866i)T \) |
| 23 | \( 1 + (0.733 + 0.680i)T \) |
| 29 | \( 1 + (-0.955 - 0.294i)T \) |
| 31 | \( 1 - T \) |
| 37 | \( 1 + (-0.733 + 0.680i)T \) |
| 41 | \( 1 + (0.826 + 0.563i)T \) |
| 43 | \( 1 + (0.826 - 0.563i)T \) |
| 47 | \( 1 + (0.623 + 0.781i)T \) |
| 53 | \( 1 + (0.733 + 0.680i)T \) |
| 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 | \( 1 + (-0.365 - 0.930i)T \) |
| 79 | \( 1 + T \) |
| 83 | \( 1 + (0.365 + 0.930i)T \) |
| 89 | \( 1 + (-0.988 - 0.149i)T \) |
| 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
−24.26992965830982010872229489543, −23.21045419223856084132976143873, −22.58351715776532817734357546189, −21.68404646449272440913217635994, −20.459172659248761840610378690202, −19.42426802402079212890592712230, −18.69665771164193717220188405514, −18.078838663296182220510492829183, −17.18186987834487067223498149985, −16.23228830355047441270789055502, −15.4025856606119415793139974012, −14.5770015175808100172213259517, −13.9022783169875324126815485114, −12.801074460471389068769097251563, −11.2071997502636769319553039372, −10.681751902972284981011822376213, −9.72871656429521363029963464191, −8.74258957462740703763073848531, −7.59942588063317053224691455925, −7.09195692232497536263007079378, −5.82969633136397142861729557982, −5.23977333136919136879815428138, −3.50088583298407011829096724661, −2.416873609169058271214472308, −0.6559649600721682271464731129,
1.26113580945228031370945678044, 2.097809174415984498393873691518, 3.57589134428681753944571594415, 4.53677602721146803281574261630, 5.57423054482012498020551921387, 7.293321530280584423732811814641, 7.91218039143794465141226984054, 9.196361290737281779537975664193, 9.57530272006005139514323784357, 10.651804491254961381559542947205, 11.85835154503879531598650912796, 12.391088725124507902210336594717, 13.203788423096498892051440666, 14.27379308168680029645277639397, 15.591030090160716827385890386756, 16.68742188813594913584481821430, 17.03758022483934208932160653058, 18.12170197255497000737012277647, 18.9909192614246663679310876745, 19.785602006682256099584412567865, 20.80416503683029096772345866238, 21.010537187143434820359200348617, 22.15003646406904768649814994653, 23.15170599794437636201828976064, 24.06274297246464651741611047320