L(s) = 1 | + (−0.5 + 0.866i)3-s + (0.5 − 0.866i)5-s − 7-s + (−0.5 − 0.866i)9-s + 11-s + (0.5 + 0.866i)13-s + (0.5 + 0.866i)15-s + (−0.5 + 0.866i)17-s + (0.5 − 0.866i)21-s + (0.5 + 0.866i)23-s + (−0.5 − 0.866i)25-s + 27-s + (0.5 + 0.866i)29-s − 31-s + (−0.5 + 0.866i)33-s + ⋯ |
L(s) = 1 | + (−0.5 + 0.866i)3-s + (0.5 − 0.866i)5-s − 7-s + (−0.5 − 0.866i)9-s + 11-s + (0.5 + 0.866i)13-s + (0.5 + 0.866i)15-s + (−0.5 + 0.866i)17-s + (0.5 − 0.866i)21-s + (0.5 + 0.866i)23-s + (−0.5 − 0.866i)25-s + 27-s + (0.5 + 0.866i)29-s − 31-s + (−0.5 + 0.866i)33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.305 + 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.305 + 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6546393926 + 0.8978072312i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6546393926 + 0.8978072312i\) |
\(L(1)\) |
\(\approx\) |
\(0.8404346409 + 0.2699053707i\) |
\(L(1)\) |
\(\approx\) |
\(0.8404346409 + 0.2699053707i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 \) |
good | 3 | \( 1 + (-0.5 + 0.866i)T \) |
| 5 | \( 1 + (0.5 - 0.866i)T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + (0.5 + 0.866i)T \) |
| 17 | \( 1 + (-0.5 + 0.866i)T \) |
| 23 | \( 1 + (0.5 + 0.866i)T \) |
| 29 | \( 1 + (0.5 + 0.866i)T \) |
| 31 | \( 1 - T \) |
| 37 | \( 1 - T \) |
| 41 | \( 1 + (-0.5 + 0.866i)T \) |
| 43 | \( 1 + (-0.5 + 0.866i)T \) |
| 47 | \( 1 + (0.5 + 0.866i)T \) |
| 53 | \( 1 + (0.5 + 0.866i)T \) |
| 59 | \( 1 + (-0.5 + 0.866i)T \) |
| 61 | \( 1 + (0.5 + 0.866i)T \) |
| 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 | \( 1 + T \) |
| 89 | \( 1 + (-0.5 - 0.866i)T \) |
| 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
−27.583016462216010341620369988910, −26.40299733358719653884143581911, −25.23750723109587970063652019848, −24.90776311915742593976435419913, −23.35039009799455609393005912090, −22.49585464017863927510466997751, −22.15867373591211527990659044358, −20.40333888255048125983546468732, −19.299100640154004212433587226741, −18.54238525076061990450483573321, −17.62770844808576588795315136252, −16.69585672575468443432119137095, −15.42267994588746653568416592711, −14.056908632268930651648144186302, −13.29851105553702268767248099091, −12.22134099107134383717519706368, −11.08918677862201887280474304211, −10.08293854605747487592896982653, −8.742920710928747263199605413554, −7.079726973088655034888089480183, −6.54592414214426878214092212596, −5.48081472186279465291754994642, −3.44557847114345774774327111419, −2.20208646241402772319265226454, −0.473583690225001973865932685105,
1.3650927250625692420893540971, 3.491800122126944042220834625768, 4.50561691194787151793307987885, 5.83331846201007071478383458744, 6.66200777806023815913407706983, 8.90030275353025450796485732112, 9.2902766297732917579655450400, 10.48166027773724319977399367863, 11.72306955300970982763877694166, 12.7327877700280747706532352546, 13.876210897634160209824950413607, 15.19751818734286401585872609526, 16.33522300127354192024986990512, 16.79278498381389134423974663422, 17.817994710858713387236800966004, 19.400881083340813757670435597436, 20.22828527280959979975628266993, 21.45914270356078081830928827651, 21.948887620679708352192514290002, 23.11674064409503310492322030938, 24.05332246268481185163285162085, 25.37158634238380915586504729349, 26.0677009997664327295389456771, 27.27099473190211334392124129419, 28.190233849836519985119902007506