L(s) = 1 | + (0.5 − 0.866i)2-s + 3-s + (−0.5 − 0.866i)4-s + (−0.5 + 0.866i)5-s + (0.5 − 0.866i)6-s + (0.5 − 0.866i)7-s − 8-s + 9-s + (0.5 + 0.866i)10-s − 11-s + (−0.5 − 0.866i)12-s + (−0.5 + 0.866i)13-s + (−0.5 − 0.866i)14-s + (−0.5 + 0.866i)15-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)17-s + ⋯ |
L(s) = 1 | + (0.5 − 0.866i)2-s + 3-s + (−0.5 − 0.866i)4-s + (−0.5 + 0.866i)5-s + (0.5 − 0.866i)6-s + (0.5 − 0.866i)7-s − 8-s + 9-s + (0.5 + 0.866i)10-s − 11-s + (−0.5 − 0.866i)12-s + (−0.5 + 0.866i)13-s + (−0.5 − 0.866i)14-s + (−0.5 + 0.866i)15-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.462 - 0.886i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.462 - 0.886i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.131383620 - 0.6858246408i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.131383620 - 0.6858246408i\) |
\(L(1)\) |
\(\approx\) |
\(1.306892729 - 0.5799865204i\) |
\(L(1)\) |
\(\approx\) |
\(1.306892729 - 0.5799865204i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 61 | \( 1 \) |
good | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + (-0.5 + 0.866i)T \) |
| 7 | \( 1 + (0.5 - 0.866i)T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + (-0.5 + 0.866i)T \) |
| 17 | \( 1 + (0.5 + 0.866i)T \) |
| 19 | \( 1 + (-0.5 - 0.866i)T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + (0.5 + 0.866i)T \) |
| 31 | \( 1 + (0.5 + 0.866i)T \) |
| 37 | \( 1 - T \) |
| 41 | \( 1 + T \) |
| 43 | \( 1 + (0.5 - 0.866i)T \) |
| 47 | \( 1 + (-0.5 - 0.866i)T \) |
| 53 | \( 1 - T \) |
| 59 | \( 1 + (0.5 - 0.866i)T \) |
| 67 | \( 1 + (0.5 - 0.866i)T \) |
| 71 | \( 1 + (0.5 + 0.866i)T \) |
| 73 | \( 1 + (-0.5 - 0.866i)T \) |
| 79 | \( 1 + (0.5 - 0.866i)T \) |
| 83 | \( 1 + (-0.5 + 0.866i)T \) |
| 89 | \( 1 - T \) |
| 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
−32.332397028582635743087533094439, −31.69199747752425351313728087761, −31.15841736469482571926864787082, −29.79926602129773772648480728535, −27.871520952546668105657475333768, −27.03626754340121736588527385870, −25.73026641846363910407617459322, −24.76391899438881796169598966447, −24.18324589175117322012554295721, −22.80608764120406999440779599212, −21.20773314214365924266523863264, −20.63722767772497200963889580612, −18.98174199547977574460303159874, −17.758054460919639425331223906906, −16.06934909591792193919298620008, −15.40798519804130329586156830211, −14.29205677048807924540351397186, −12.960337789352319690870473592775, −12.11566587734412236225392939208, −9.6099302065634368802028203685, −8.16035313439767258608874297113, −7.84331161013990297408994327117, −5.57182733280192471675925063704, −4.38539595882684609887470446619, −2.71090722267694431553152586772,
2.082633829785443739605837851793, 3.44392883425702581454096118525, 4.58625260794530546378074000939, 6.94285558532070839428565400323, 8.3172854906401107318267871354, 10.06992794730389083961293709993, 10.87191789262440677802401538174, 12.44359544768180114243884536944, 13.853029705849148073012538496834, 14.48838806802647004420570464831, 15.6553689182284219110761306530, 17.90693020071791309112458452600, 19.11245994943717360535158892025, 19.79056424652425608898750703856, 21.01330022864515481041508845981, 21.85807737601701341013502376276, 23.50537292645569377201249970082, 24.00939572750078286398961065942, 26.07302774612770157783581227490, 26.69577594244334007532406368691, 27.86479327911104326752742480727, 29.48129004384622402565696422515, 30.357175281733102095930080466147, 31.03876793177522121793180326945, 31.99188116392148569652143425592