L(s) = 1 | + (0.365 − 0.930i)3-s + (−0.988 + 0.149i)5-s + (−0.733 − 0.680i)9-s + (0.733 − 0.680i)11-s + (−0.222 − 0.974i)13-s + (−0.222 + 0.974i)15-s + (−0.0747 − 0.997i)17-s + (−0.5 + 0.866i)19-s + (0.0747 − 0.997i)23-s + (0.955 − 0.294i)25-s + (−0.900 + 0.433i)27-s + (0.900 + 0.433i)29-s + (0.5 + 0.866i)31-s + (−0.365 − 0.930i)33-s + (−0.826 + 0.563i)37-s + ⋯ |
L(s) = 1 | + (0.365 − 0.930i)3-s + (−0.988 + 0.149i)5-s + (−0.733 − 0.680i)9-s + (0.733 − 0.680i)11-s + (−0.222 − 0.974i)13-s + (−0.222 + 0.974i)15-s + (−0.0747 − 0.997i)17-s + (−0.5 + 0.866i)19-s + (0.0747 − 0.997i)23-s + (0.955 − 0.294i)25-s + (−0.900 + 0.433i)27-s + (0.900 + 0.433i)29-s + (0.5 + 0.866i)31-s + (−0.365 − 0.930i)33-s + (−0.826 + 0.563i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.695 + 0.718i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.695 + 0.718i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.1827446655 - 0.4313735935i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.1827446655 - 0.4313735935i\) |
\(L(1)\) |
\(\approx\) |
\(0.7544452710 - 0.3818649094i\) |
\(L(1)\) |
\(\approx\) |
\(0.7544452710 - 0.3818649094i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (0.365 - 0.930i)T \) |
| 5 | \( 1 + (-0.988 + 0.149i)T \) |
| 11 | \( 1 + (0.733 - 0.680i)T \) |
| 13 | \( 1 + (-0.222 - 0.974i)T \) |
| 17 | \( 1 + (-0.0747 - 0.997i)T \) |
| 19 | \( 1 + (-0.5 + 0.866i)T \) |
| 23 | \( 1 + (0.0747 - 0.997i)T \) |
| 29 | \( 1 + (0.900 + 0.433i)T \) |
| 31 | \( 1 + (0.5 + 0.866i)T \) |
| 37 | \( 1 + (-0.826 + 0.563i)T \) |
| 41 | \( 1 + (-0.623 - 0.781i)T \) |
| 43 | \( 1 + (-0.623 + 0.781i)T \) |
| 47 | \( 1 + (-0.955 - 0.294i)T \) |
| 53 | \( 1 + (-0.826 - 0.563i)T \) |
| 59 | \( 1 + (-0.988 - 0.149i)T \) |
| 61 | \( 1 + (0.826 - 0.563i)T \) |
| 67 | \( 1 + (0.5 + 0.866i)T \) |
| 71 | \( 1 + (-0.900 + 0.433i)T \) |
| 73 | \( 1 + (-0.955 + 0.294i)T \) |
| 79 | \( 1 + (-0.5 + 0.866i)T \) |
| 83 | \( 1 + (-0.222 + 0.974i)T \) |
| 89 | \( 1 + (0.733 + 0.680i)T \) |
| 97 | \( 1 - T \) |
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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
−24.81867748792799646936413099485, −23.77831672360978073884044070349, −23.0483086322322581290617026450, −22.03148610062437754737075145883, −21.37859333600763923900936970662, −20.345959600979960236215836638646, −19.525388833335894953483936255348, −19.154883711133879297023743547177, −17.43992392833067329457352652162, −16.82504586148327408643949165318, −15.75455403592319781534101865289, −15.1620407806076599690953681058, −14.41364316898091192176187207013, −13.27274164465785700495922059825, −11.979870912736746973785204993255, −11.37303214765820052963228682511, −10.2804800946394750076725134335, −9.26677273757745167709240632800, −8.542941457200768371676944621352, −7.48523452372599068713161830703, −6.358490496831730620580038181377, −4.75530157837189662399993657262, −4.2375447367217243911095526118, −3.26034800204770847057504308138, −1.81219043941298142945219458802,
0.13057122950604092307072011694, 1.20707043560455222582158515104, 2.81323868605646931536406369070, 3.534575480544481868294997873704, 4.957506594444196135357586449278, 6.3781576181833420169715528508, 7.07595977182657052135248455285, 8.245588602032002837778083922937, 8.598356907429478927425667038508, 10.14756485841558724933321898367, 11.31844078125154960764488664263, 12.1071553275323161698000341170, 12.7871841423164131999557024362, 14.02963261745952742099171379841, 14.60681532354011870443415633468, 15.67651886794458264401481619062, 16.6416470387496379369062325636, 17.71103926419546971337722379007, 18.63193409399121338540510374619, 19.275516867027320971391910267092, 20.03691462779591019619006406166, 20.76737985220688597007857409755, 22.24163261718848855469568252286, 22.95814853287500026329006274575, 23.66540778926706980302272027845