L(s) = 1 | + (0.994 + 0.104i)3-s + (0.978 + 0.207i)9-s + (−0.978 + 0.207i)11-s + (−0.951 + 0.309i)13-s + (−0.406 + 0.913i)17-s + (0.104 + 0.994i)19-s + (−0.743 + 0.669i)23-s + (0.951 + 0.309i)27-s + (−0.809 − 0.587i)29-s + (−0.913 − 0.406i)31-s + (−0.994 + 0.104i)33-s + (−0.207 + 0.978i)37-s + (−0.978 + 0.207i)39-s + (0.309 + 0.951i)41-s − i·43-s + ⋯ |
L(s) = 1 | + (0.994 + 0.104i)3-s + (0.978 + 0.207i)9-s + (−0.978 + 0.207i)11-s + (−0.951 + 0.309i)13-s + (−0.406 + 0.913i)17-s + (0.104 + 0.994i)19-s + (−0.743 + 0.669i)23-s + (0.951 + 0.309i)27-s + (−0.809 − 0.587i)29-s + (−0.913 − 0.406i)31-s + (−0.994 + 0.104i)33-s + (−0.207 + 0.978i)37-s + (−0.978 + 0.207i)39-s + (0.309 + 0.951i)41-s − i·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.637 + 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.637 + 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4915669005 + 1.043896612i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4915669005 + 1.043896612i\) |
\(L(1)\) |
\(\approx\) |
\(1.104207682 + 0.2868901200i\) |
\(L(1)\) |
\(\approx\) |
\(1.104207682 + 0.2868901200i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (0.994 + 0.104i)T \) |
| 11 | \( 1 + (-0.978 + 0.207i)T \) |
| 13 | \( 1 + (-0.951 + 0.309i)T \) |
| 17 | \( 1 + (-0.406 + 0.913i)T \) |
| 19 | \( 1 + (0.104 + 0.994i)T \) |
| 23 | \( 1 + (-0.743 + 0.669i)T \) |
| 29 | \( 1 + (-0.809 - 0.587i)T \) |
| 31 | \( 1 + (-0.913 - 0.406i)T \) |
| 37 | \( 1 + (-0.207 + 0.978i)T \) |
| 41 | \( 1 + (0.309 + 0.951i)T \) |
| 43 | \( 1 - iT \) |
| 47 | \( 1 + (-0.406 - 0.913i)T \) |
| 53 | \( 1 + (-0.994 - 0.104i)T \) |
| 59 | \( 1 + (-0.669 + 0.743i)T \) |
| 61 | \( 1 + (-0.669 - 0.743i)T \) |
| 67 | \( 1 + (-0.406 + 0.913i)T \) |
| 71 | \( 1 + (0.809 + 0.587i)T \) |
| 73 | \( 1 + (0.207 + 0.978i)T \) |
| 79 | \( 1 + (0.913 - 0.406i)T \) |
| 83 | \( 1 + (0.587 + 0.809i)T \) |
| 89 | \( 1 + (-0.669 - 0.743i)T \) |
| 97 | \( 1 + (0.587 - 0.809i)T \) |
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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
−20.43433781441016649925740988041, −19.82531672187019774208173290278, −19.16997392803103455153755483296, −18.14789456412112328136554232784, −17.90324352681230165770382422094, −16.54983432508693963104322431346, −15.848543945013770212594290482234, −15.2044053852792584218068581820, −14.34489575638081273327542747918, −13.776220814397244384209666778615, −12.8417243309047788514444808818, −12.42281779892821571675585491813, −11.12021810601889201359620615037, −10.43892304523378387389410093973, −9.39733084960814491161604752662, −9.005761755870804063604225557660, −7.777837131679175326802898003241, −7.49401985571211997254631370916, −6.476496485251452696852707261602, −5.19802448833272715199602667209, −4.57263702737919646524127935590, −3.3603641105017094403966286161, −2.6438628220643179565328010227, −1.9166010756174796837436422067, −0.33705842623251895775984344237,
1.706639109365546820957107454751, 2.25729918814022596674499857074, 3.35513667685768187882510266462, 4.120402012827152399158756208152, 5.0482717509707049227263941837, 6.04239456018199742554702751534, 7.21265731745453274796919183474, 7.835733845455575854791969416452, 8.46398526418442692703108947573, 9.58875154275394410434726587633, 9.98455954888174447788297398462, 10.888737337116148302291311516463, 12.01890207669125221984604811660, 12.81205505262231319903180835016, 13.441582981634000500273877711631, 14.28637032829956255831933123300, 15.020357491290448883487150721129, 15.52709341477978366462102716925, 16.47763064949599700752165669823, 17.24582551439020155074191222253, 18.32386180719860041617195279462, 18.80748020176622999135656525155, 19.68906222039843513512349242351, 20.25043476890884794042907086583, 21.00675472639682783403388250041