L(s) = 1 | + (0.929 − 0.368i)3-s + (−0.535 + 0.844i)5-s + (−0.929 + 0.368i)7-s + (0.728 − 0.684i)9-s + (0.809 + 0.587i)11-s + (0.992 + 0.125i)13-s + (−0.187 + 0.982i)15-s + (0.0627 − 0.998i)17-s + (−0.876 + 0.481i)19-s + (−0.728 + 0.684i)21-s + (0.876 + 0.481i)23-s + (−0.425 − 0.904i)25-s + (0.425 − 0.904i)27-s + (−0.728 + 0.684i)29-s + (0.728 − 0.684i)31-s + ⋯ |
L(s) = 1 | + (0.929 − 0.368i)3-s + (−0.535 + 0.844i)5-s + (−0.929 + 0.368i)7-s + (0.728 − 0.684i)9-s + (0.809 + 0.587i)11-s + (0.992 + 0.125i)13-s + (−0.187 + 0.982i)15-s + (0.0627 − 0.998i)17-s + (−0.876 + 0.481i)19-s + (−0.728 + 0.684i)21-s + (0.876 + 0.481i)23-s + (−0.425 − 0.904i)25-s + (0.425 − 0.904i)27-s + (−0.728 + 0.684i)29-s + (0.728 − 0.684i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.105i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.105i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.364638957 + 0.1254246887i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.364638957 + 0.1254246887i\) |
\(L(1)\) |
\(\approx\) |
\(1.359486030 + 0.05889468143i\) |
\(L(1)\) |
\(\approx\) |
\(1.359486030 + 0.05889468143i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 751 | \( 1 \) |
good | 3 | \( 1 + (0.929 - 0.368i)T \) |
| 5 | \( 1 + (-0.535 + 0.844i)T \) |
| 7 | \( 1 + (-0.929 + 0.368i)T \) |
| 11 | \( 1 + (0.809 + 0.587i)T \) |
| 13 | \( 1 + (0.992 + 0.125i)T \) |
| 17 | \( 1 + (0.0627 - 0.998i)T \) |
| 19 | \( 1 + (-0.876 + 0.481i)T \) |
| 23 | \( 1 + (0.876 + 0.481i)T \) |
| 29 | \( 1 + (-0.728 + 0.684i)T \) |
| 31 | \( 1 + (0.728 - 0.684i)T \) |
| 37 | \( 1 + (0.425 + 0.904i)T \) |
| 41 | \( 1 + (-0.809 - 0.587i)T \) |
| 43 | \( 1 + (-0.535 - 0.844i)T \) |
| 47 | \( 1 + (0.535 - 0.844i)T \) |
| 53 | \( 1 + (0.809 - 0.587i)T \) |
| 59 | \( 1 + (0.637 + 0.770i)T \) |
| 61 | \( 1 + (0.809 - 0.587i)T \) |
| 67 | \( 1 + (-0.728 + 0.684i)T \) |
| 71 | \( 1 + (-0.187 - 0.982i)T \) |
| 73 | \( 1 + T \) |
| 79 | \( 1 + (0.876 + 0.481i)T \) |
| 83 | \( 1 + (-0.309 - 0.951i)T \) |
| 89 | \( 1 + (0.535 - 0.844i)T \) |
| 97 | \( 1 + (-0.929 - 0.368i)T \) |
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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
−17.514767451121818398744621078664, −16.71114488005845782579484470446, −16.482858377049281503687925246539, −15.683484581613753556560816491107, −15.191555794300552823411298813425, −14.50832539570370147401982807955, −13.624152198871493349646496484488, −13.08673975389858629617413584101, −12.78142374155559928363790736383, −11.80256262091884069724458470995, −10.906842175864896353400272583732, −10.47224353016741960689879954186, −9.38962876219479105610703399159, −9.10344464951084768483066060004, −8.34774269449272692917545574528, −7.99280129154726185466937746262, −6.86655738263389218801785529607, −6.35772891322312564828626090030, −5.423755821540018176350358137023, −4.362257210258451520433125939527, −3.960409503184641445616678427609, −3.4225923622700536683519276772, −2.59344195363392426877000531096, −1.463859826500663137025148771960, −0.79010627096484769337099992275,
0.71722841241656279570377493591, 1.837726684549533206939920442348, 2.48662480354815770221198421829, 3.436990149023501578541975724568, 3.600230370261420250037782798081, 4.49144981855115871535683622067, 5.71921034430567869607537035602, 6.60333743951877863280705320943, 6.89350492927047881562169998753, 7.51840844814329917336509832540, 8.55891909623016072076425090917, 8.87027479436917925886301111417, 9.77839516039181431759722717246, 10.1996847474021987973436553088, 11.24508231012990524616682230515, 11.881574173691444586603051529545, 12.43595826720664059535564370303, 13.38947592294328112686658284978, 13.62769614474672831220853663459, 14.60005910090853993026613806211, 15.12852543125445555693410253600, 15.46105747606385635878037213998, 16.32003432270302628378452046711, 17.007765346651547639810779526366, 18.047287370425667643323758781246