L(s) = 1 | + (0.809 − 0.587i)2-s + (0.309 + 0.951i)3-s + (0.309 − 0.951i)4-s + (−0.809 − 0.587i)5-s + (0.809 + 0.587i)6-s + (−0.309 + 0.951i)7-s + (−0.309 − 0.951i)8-s + (−0.809 + 0.587i)9-s − 10-s + 12-s + (0.809 − 0.587i)13-s + (0.309 + 0.951i)14-s + (0.309 − 0.951i)15-s + (−0.809 − 0.587i)16-s + (0.809 + 0.587i)17-s + (−0.309 + 0.951i)18-s + ⋯ |
L(s) = 1 | + (0.809 − 0.587i)2-s + (0.309 + 0.951i)3-s + (0.309 − 0.951i)4-s + (−0.809 − 0.587i)5-s + (0.809 + 0.587i)6-s + (−0.309 + 0.951i)7-s + (−0.309 − 0.951i)8-s + (−0.809 + 0.587i)9-s − 10-s + 12-s + (0.809 − 0.587i)13-s + (0.309 + 0.951i)14-s + (0.309 − 0.951i)15-s + (−0.809 − 0.587i)16-s + (0.809 + 0.587i)17-s + (−0.309 + 0.951i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.957 - 0.288i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.957 - 0.288i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.503380081 - 0.2211997697i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.503380081 - 0.2211997697i\) |
\(L(1)\) |
\(\approx\) |
\(1.415747663 - 0.1740328837i\) |
\(L(1)\) |
\(\approx\) |
\(1.415747663 - 0.1740328837i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 \) |
good | 2 | \( 1 + (0.809 - 0.587i)T \) |
| 3 | \( 1 + (0.309 + 0.951i)T \) |
| 5 | \( 1 + (-0.809 - 0.587i)T \) |
| 7 | \( 1 + (-0.309 + 0.951i)T \) |
| 13 | \( 1 + (0.809 - 0.587i)T \) |
| 17 | \( 1 + (0.809 + 0.587i)T \) |
| 19 | \( 1 + (-0.309 - 0.951i)T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + (-0.309 + 0.951i)T \) |
| 31 | \( 1 + (-0.809 + 0.587i)T \) |
| 37 | \( 1 + (0.309 - 0.951i)T \) |
| 41 | \( 1 + (-0.309 - 0.951i)T \) |
| 43 | \( 1 - T \) |
| 47 | \( 1 + (0.309 + 0.951i)T \) |
| 53 | \( 1 + (-0.809 + 0.587i)T \) |
| 59 | \( 1 + (0.309 - 0.951i)T \) |
| 61 | \( 1 + (0.809 + 0.587i)T \) |
| 67 | \( 1 + T \) |
| 71 | \( 1 + (-0.809 - 0.587i)T \) |
| 73 | \( 1 + (-0.309 + 0.951i)T \) |
| 79 | \( 1 + (0.809 - 0.587i)T \) |
| 83 | \( 1 + (0.809 + 0.587i)T \) |
| 89 | \( 1 + T \) |
| 97 | \( 1 + (-0.809 + 0.587i)T \) |
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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
−45.571883708328470662009967870960, −43.13903733098448016886778132619, −42.601218189354830328691114239797, −41.34346950295061398221815956035, −39.91816544484692915524132036871, −38.421569250426374648790601150719, −36.103221852765725499410258250258, −35.0748181390130967490481952242, −33.60918441512861464269747264056, −31.76704292714201836153420720265, −30.615315398535949332903421061066, −29.55920151127941976080946946763, −26.67028496809013083238174485224, −25.38976369113433935050441838210, −23.62369920986579610417564305629, −22.95929393378876961523787194156, −20.50797618286869986017769960585, −18.7495375665351457827169284855, −16.64292363517938135933695505995, −14.674032161018154187712356107968, −13.33022743788218395645836666989, −11.59406426254877199887647483811, −7.88643465920150753487622877767, −6.63073045048494212311693469779, −3.54704109171945007666447637176,
3.41492187932220368062022478551, 5.27308657584208989337435824912, 8.95354546437232036116473143789, 11.00919009252031218454364175306, 12.73163403755865549814505940274, 14.94827047321579871059739778452, 16.003353712811180942448748166867, 19.23890660918179306832270433212, 20.5790557747237967713122828270, 21.88489757248339392930656813056, 23.38804121013922687721723923210, 25.294856636069449525514142678791, 27.61231692090569895478179228700, 28.4508199966808353191563583286, 30.758854343797428801601385292783, 31.87403308116394856064712507976, 32.799985112149809068419874207202, 34.72345992628301281730177232071, 37.01755954390176887573149124687, 38.395494452428109122979398955656, 39.22705738819166212922797709288, 40.66058066592356276517670187998, 42.45917275651500585448292842681, 43.63192017509203526976799880310, 45.30426327898638018804634367866