L(s) = 1 | − i·3-s − 9-s + 11-s + i·13-s + i·17-s + 19-s − i·23-s + i·27-s + 29-s + 31-s − i·33-s + i·37-s + 39-s − 41-s + i·43-s + ⋯ |
L(s) = 1 | − i·3-s − 9-s + 11-s + i·13-s + i·17-s + 19-s − i·23-s + i·27-s + 29-s + 31-s − i·33-s + i·37-s + 39-s − 41-s + i·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.934154282 - 0.5494526991i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.934154282 - 0.5494526991i\) |
\(L(1)\) |
\(\approx\) |
\(1.155645983 - 0.2728110099i\) |
\(L(1)\) |
\(\approx\) |
\(1.155645983 - 0.2728110099i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + T \) |
| 11 | \( 1 \) |
| 13 | \( 1 \) |
| 17 | \( 1 \) |
| 19 | \( 1 \) |
| 23 | \( 1 \) |
| 29 | \( 1 - T \) |
| 31 | \( 1 \) |
| 37 | \( 1 + T \) |
| 41 | \( 1 \) |
| 43 | \( 1 + iT \) |
| 47 | \( 1 \) |
| 53 | \( 1 \) |
| 59 | \( 1 \) |
| 61 | \( 1 + iT \) |
| 67 | \( 1 \) |
| 71 | \( 1 + T \) |
| 73 | \( 1 \) |
| 79 | \( 1 \) |
| 83 | \( 1 \) |
| 89 | \( 1 - iT \) |
| 97 | \( 1 \) |
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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
−25.32259365909469065874465892642, −24.918511881410669399775988143910, −23.44433589961662483199081283875, −22.52405383697051905999657273905, −22.02000263463835300610196498473, −20.87191050301354501587310925933, −20.16380591027998232146705775090, −19.339242026262611498905598011045, −17.88075367454636524601180533128, −17.22026107911893479535551203811, −16.05907058672447984033824855419, −15.49014013685939208809523704351, −14.371929771375713067488348454332, −13.632236819265135161397759709871, −12.08203600813524035828673293524, −11.37601962583367210249377869410, −10.191899154795777553507665679621, −9.46275281374642124718159270262, −8.47386857961626936015738858543, −7.21485375698677417549998274613, −5.82916618955331821622567162857, −4.933919164296949052242288548893, −3.74089204685326030688819066473, −2.79270414583279243473136719958, −0.85078110766109696022108555889,
0.95569305997409724696479945271, 2.01119611640031755632030253667, 3.40421019924881953641296943571, 4.77685234916196291293364539590, 6.301873112321556336602027458599, 6.76544203306655477957835358138, 8.09905496985459624398258368570, 8.90093614883574278590170642302, 10.163249858519506700815874100577, 11.60802453979509647449252299604, 12.03217215079330589271815122418, 13.233854056077717673478585966229, 14.08825605204932107298335967021, 14.85053490926988280985022945947, 16.3398073705486419295453291419, 17.1340318611367464043870308109, 18.02442600901318977704991291824, 19.02966820330244711799835422362, 19.6114985308904206884142394917, 20.62170320555495611829387812975, 21.845232174939884646477734304995, 22.67956717282325962712250125695, 23.68027601455735690779991715054, 24.40707173051186116051909832375, 25.13400552669562073866701795883