L(s) = 1 | + (0.448 + 0.893i)2-s + (−0.0878 − 0.996i)3-s + (−0.597 + 0.801i)4-s + (−0.332 − 0.942i)5-s + (0.850 − 0.525i)6-s + (0.656 + 0.754i)7-s + (−0.984 − 0.175i)8-s + (−0.984 + 0.175i)9-s + (0.693 − 0.720i)10-s + (0.0627 + 0.998i)11-s + (0.850 + 0.525i)12-s + (0.762 − 0.647i)13-s + (−0.379 + 0.925i)14-s + (−0.910 + 0.414i)15-s + (−0.285 − 0.958i)16-s + (0.617 + 0.786i)17-s + ⋯ |
L(s) = 1 | + (0.448 + 0.893i)2-s + (−0.0878 − 0.996i)3-s + (−0.597 + 0.801i)4-s + (−0.332 − 0.942i)5-s + (0.850 − 0.525i)6-s + (0.656 + 0.754i)7-s + (−0.984 − 0.175i)8-s + (−0.984 + 0.175i)9-s + (0.693 − 0.720i)10-s + (0.0627 + 0.998i)11-s + (0.850 + 0.525i)12-s + (0.762 − 0.647i)13-s + (−0.379 + 0.925i)14-s + (−0.910 + 0.414i)15-s + (−0.285 − 0.958i)16-s + (0.617 + 0.786i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 751 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.443 + 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 751 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.443 + 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.274837701 + 0.7919758545i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.274837701 + 0.7919758545i\) |
\(L(1)\) |
\(\approx\) |
\(1.140927892 + 0.3259220596i\) |
\(L(1)\) |
\(\approx\) |
\(1.140927892 + 0.3259220596i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 751 | \( 1 \) |
good | 2 | \( 1 + (0.448 + 0.893i)T \) |
| 3 | \( 1 + (-0.0878 - 0.996i)T \) |
| 5 | \( 1 + (-0.332 - 0.942i)T \) |
| 7 | \( 1 + (0.656 + 0.754i)T \) |
| 11 | \( 1 + (0.0627 + 0.998i)T \) |
| 13 | \( 1 + (0.762 - 0.647i)T \) |
| 17 | \( 1 + (0.617 + 0.786i)T \) |
| 19 | \( 1 + (0.0125 + 0.999i)T \) |
| 23 | \( 1 + (-0.947 - 0.320i)T \) |
| 29 | \( 1 + (-0.470 - 0.882i)T \) |
| 31 | \( 1 + (0.693 - 0.720i)T \) |
| 37 | \( 1 + (0.356 + 0.934i)T \) |
| 41 | \( 1 + (0.0627 + 0.998i)T \) |
| 43 | \( 1 + (0.823 - 0.567i)T \) |
| 47 | \( 1 + (-0.285 + 0.958i)T \) |
| 53 | \( 1 + (0.535 + 0.844i)T \) |
| 59 | \( 1 + (0.402 - 0.915i)T \) |
| 61 | \( 1 + (0.535 + 0.844i)T \) |
| 67 | \( 1 + (0.899 + 0.437i)T \) |
| 71 | \( 1 + (-0.675 + 0.737i)T \) |
| 73 | \( 1 + T \) |
| 79 | \( 1 + (0.0125 - 0.999i)T \) |
| 83 | \( 1 + (0.728 - 0.684i)T \) |
| 89 | \( 1 + (0.823 + 0.567i)T \) |
| 97 | \( 1 + (0.656 - 0.754i)T \) |
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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
−22.132285673360235943224204686344, −21.43577099555764656003892303273, −20.9411425497600760257073806260, −19.95902141049883343650805480702, −19.36461831700604712795365449787, −18.31363847670029845525717193740, −17.69034925775555211530361575478, −16.40553448267865459984543924235, −15.72272916345805278029705022260, −14.6676185448451872870536355487, −14.045556567367323825539868527043, −13.6475555219216090291412978980, −11.98477592217835427505839529744, −11.21234962610734070681517560631, −10.95810218158934717967463572705, −10.11202311102425372169350041131, −9.1566305707891678640486207621, −8.23514982347895566360055962904, −6.88243594893374791166269692833, −5.7972292918787977387749887244, −4.88513307915141945767054253496, −3.827490290325431567526509095082, −3.462578868431339972547842895198, −2.31441572182960904566851428808, −0.725245350753815353189705482711,
1.16699982625482010267564314684, 2.30946005998214310977216022051, 3.7792653414838418277232937952, 4.734313299906222004573885928871, 5.78212720063237125732023148579, 6.13793890281852450023919146109, 7.69487439674694705551672615822, 8.00085549895926261311986161339, 8.6554557671960383763067927758, 9.836861722505448414736694168394, 11.52532007846310911986234272856, 12.19536787645300611760359111543, 12.70400064920445920855779535483, 13.46593536404901413527450013492, 14.51149613210714702530716255012, 15.15907365127839147609800432921, 16.04573675055911572613565949884, 17.029558322932075747101685244486, 17.56753928338018207036319373811, 18.37244804617745753744607086629, 19.10506426458099138276491917612, 20.50217837897967981503699674502, 20.7694172496945073036735519790, 22.04862157726157333890809096273, 22.9566711242831543642053743612