L(s) = 1 | + (0.309 − 0.951i)3-s + (0.809 − 0.587i)5-s + (−0.309 − 0.951i)7-s + (−0.809 − 0.587i)9-s + (0.809 + 0.587i)13-s + (−0.309 − 0.951i)15-s + (−0.809 + 0.587i)17-s + (0.309 − 0.951i)19-s − 21-s − 23-s + (0.309 − 0.951i)25-s + (−0.809 + 0.587i)27-s + (−0.309 − 0.951i)29-s + (0.809 + 0.587i)31-s + (−0.809 − 0.587i)35-s + ⋯ |
L(s) = 1 | + (0.309 − 0.951i)3-s + (0.809 − 0.587i)5-s + (−0.309 − 0.951i)7-s + (−0.809 − 0.587i)9-s + (0.809 + 0.587i)13-s + (−0.309 − 0.951i)15-s + (−0.809 + 0.587i)17-s + (0.309 − 0.951i)19-s − 21-s − 23-s + (0.309 − 0.951i)25-s + (−0.809 + 0.587i)27-s + (−0.309 − 0.951i)29-s + (0.809 + 0.587i)31-s + (−0.809 − 0.587i)35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.550 - 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.550 - 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8717271307 - 1.618234264i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8717271307 - 1.618234264i\) |
\(L(1)\) |
\(\approx\) |
\(1.063876633 - 0.7009158734i\) |
\(L(1)\) |
\(\approx\) |
\(1.063876633 - 0.7009158734i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 11 | \( 1 \) |
good | 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
−30.87908225465112027781009806266, −29.532722033140963358197887047282, −28.4724671947752950264511363015, −27.54688883056552055648905478846, −26.32235103870330523979051648966, −25.564965234161918392303033781443, −24.72408267276093678291664608727, −22.73618732590257298202993770701, −22.16231528941964038001468326521, −21.164443028046816871514749966767, −20.20138436907227182925347438849, −18.711279636459443899071731486205, −17.80077312048599666735229714059, −16.30349694829974112399012905400, −15.39571954484944484822939917942, −14.356378622711175153785034085202, −13.25429856662670315938774607989, −11.56600983106387843175982733375, −10.33152687561207446399690741185, −9.439070044473642036356057784760, −8.29064239039455006468524950596, −6.28991413758862863630228197953, −5.291098322690379749205319812038, −3.493119441699387204655774270311, −2.306300389137098560731572153555,
0.83183133692998259340340397727, 2.20407205182165130142839466096, 4.083778619882557491742377805612, 5.95746989830095033570418727317, 6.95131175648501722355337973554, 8.39268345715794398933695970449, 9.481510806319495884864674875994, 11.00153350922352094533657185925, 12.50669915518956772334031299391, 13.53303888003004614791877985027, 14.011155717705988705370987681255, 15.88730946534182392555298804465, 17.22103884441245449237344502630, 17.8815935218753706260705931206, 19.31399936838296020249616271329, 20.18475133814668525642067153550, 21.19973868433955540980026946276, 22.68286523707072947489309241888, 23.89624808248311798151638370917, 24.49387991526061260522257230779, 25.85889146480390728395904278025, 26.3192758954459306900119800905, 28.269885141424961700440537630318, 28.963869678393344678479471130254, 30.00192904865595353401837890430