L(s) = 1 | + (−0.354 + 0.935i)2-s + (−0.748 − 0.663i)4-s + (0.992 + 0.120i)5-s + (0.464 − 0.885i)7-s + (0.885 − 0.464i)8-s + (−0.464 + 0.885i)10-s + (0.992 − 0.120i)11-s + (−0.748 − 0.663i)13-s + (0.663 + 0.748i)14-s + (0.120 + 0.992i)16-s + (0.970 − 0.239i)19-s + (−0.663 − 0.748i)20-s + (−0.239 + 0.970i)22-s − i·23-s + (0.970 + 0.239i)25-s + (0.885 − 0.464i)26-s + ⋯ |
L(s) = 1 | + (−0.354 + 0.935i)2-s + (−0.748 − 0.663i)4-s + (0.992 + 0.120i)5-s + (0.464 − 0.885i)7-s + (0.885 − 0.464i)8-s + (−0.464 + 0.885i)10-s + (0.992 − 0.120i)11-s + (−0.748 − 0.663i)13-s + (0.663 + 0.748i)14-s + (0.120 + 0.992i)16-s + (0.970 − 0.239i)19-s + (−0.663 − 0.748i)20-s + (−0.239 + 0.970i)22-s − i·23-s + (0.970 + 0.239i)25-s + (0.885 − 0.464i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.853 - 0.521i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.853 - 0.521i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.746259851 - 0.4912163194i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.746259851 - 0.4912163194i\) |
\(L(1)\) |
\(\approx\) |
\(1.114345359 + 0.1338267838i\) |
\(L(1)\) |
\(\approx\) |
\(1.114345359 + 0.1338267838i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 17 | \( 1 \) |
| 79 | \( 1 \) |
good | 2 | \( 1 + (-0.354 + 0.935i)T \) |
| 5 | \( 1 + (0.992 + 0.120i)T \) |
| 7 | \( 1 + (0.464 - 0.885i)T \) |
| 11 | \( 1 + (0.992 - 0.120i)T \) |
| 13 | \( 1 + (-0.748 - 0.663i)T \) |
| 19 | \( 1 + (0.970 - 0.239i)T \) |
| 23 | \( 1 - iT \) |
| 29 | \( 1 + (-0.822 - 0.568i)T \) |
| 31 | \( 1 + (0.935 + 0.354i)T \) |
| 37 | \( 1 + (-0.239 - 0.970i)T \) |
| 41 | \( 1 + (0.992 + 0.120i)T \) |
| 43 | \( 1 + (0.120 - 0.992i)T \) |
| 47 | \( 1 + (-0.970 - 0.239i)T \) |
| 53 | \( 1 + (-0.885 + 0.464i)T \) |
| 59 | \( 1 + (0.748 - 0.663i)T \) |
| 61 | \( 1 + (0.239 + 0.970i)T \) |
| 67 | \( 1 + (-0.354 - 0.935i)T \) |
| 71 | \( 1 + (-0.464 - 0.885i)T \) |
| 73 | \( 1 + (0.663 + 0.748i)T \) |
| 83 | \( 1 + (-0.748 - 0.663i)T \) |
| 89 | \( 1 + (-0.885 - 0.464i)T \) |
| 97 | \( 1 + (0.239 + 0.970i)T \) |
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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
−18.535135093965694275662695341448, −17.83874524466338902285873152967, −17.421386523420830324349136557275, −16.77545439110971080291604606156, −16.02415554164943090946677326986, −14.83777993134752345706949843651, −14.31224753088117144288129018934, −13.74138516916114428814323448940, −12.88887982465755731429528174111, −12.26300446250210308631065095961, −11.53643363385241747410836580626, −11.233880503518203553568336039904, −9.909228183184217443978432452179, −9.63503523028420820159955195360, −9.10604209851218916073079177707, −8.34291884993476514768449902315, −7.48633910589763002542261280649, −6.57897523964698389413463127674, −5.63300365807347946068796052663, −4.993958433751644332339294877, −4.26150319476555029535858335886, −3.19341253433439576827618451881, −2.50825778332897589305430808612, −1.62097820389784255625963032677, −1.29653489191358303140286277133,
0.60839476735504394982942206366, 1.3394529756689631520908877508, 2.29943632878787655717619179287, 3.47974610718975597532917081896, 4.42213437940442914855790043354, 5.06486572063518441556601930076, 5.808361587588875864770082038838, 6.53906995261716128144555437465, 7.18648498479495363320514840359, 7.79170359803542570602688835127, 8.68275427845664314273407620663, 9.397690501500734585570185208196, 9.9758831116851528291469516123, 10.561118297516096960313918319918, 11.34468574217112899267180683027, 12.43795955951072001551730977917, 13.223068347414456781798656863818, 13.91318213991436380943879023195, 14.36298593923925672892412313720, 14.83361696684164201317553970755, 15.784107493730041318878785608014, 16.63263972363865235517699108103, 17.05631188815330767233426816107, 17.66062417319307718114011263580, 18.01799930724198638435123681566