| L(s) = 1 | + 1.59i·3-s − 1.60·7-s + 6.46·9-s + 2.01i·11-s + 1.88i·13-s − 1.86i·17-s + 6.62·19-s − 2.55i·21-s − 27.0i·23-s + 24.6i·27-s − 45.2i·29-s + (−13.7 − 27.7i)31-s − 3.20·33-s + 40.6i·37-s − 3.00·39-s + ⋯ |
| L(s) = 1 | + 0.530i·3-s − 0.229·7-s + 0.718·9-s + 0.183i·11-s + 0.145i·13-s − 0.109i·17-s + 0.348·19-s − 0.121i·21-s − 1.17i·23-s + 0.911i·27-s − 1.55i·29-s + (−0.443 − 0.896i)31-s − 0.0971·33-s + 1.09i·37-s − 0.0770·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.443 + 0.896i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3100 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.443 + 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.539106040\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.539106040\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 31 | \( 1 + (13.7 + 27.7i)T \) |
| good | 3 | \( 1 - 1.59iT - 9T^{2} \) |
| 7 | \( 1 + 1.60T + 49T^{2} \) |
| 11 | \( 1 - 2.01iT - 121T^{2} \) |
| 13 | \( 1 - 1.88iT - 169T^{2} \) |
| 17 | \( 1 + 1.86iT - 289T^{2} \) |
| 19 | \( 1 - 6.62T + 361T^{2} \) |
| 23 | \( 1 + 27.0iT - 529T^{2} \) |
| 29 | \( 1 + 45.2iT - 841T^{2} \) |
| 37 | \( 1 - 40.6iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 6.90T + 1.68e3T^{2} \) |
| 43 | \( 1 + 47.4iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 57.3T + 2.20e3T^{2} \) |
| 53 | \( 1 - 0.0248iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 73.5T + 3.48e3T^{2} \) |
| 61 | \( 1 - 81.6iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 50.0T + 4.48e3T^{2} \) |
| 71 | \( 1 - 77.8T + 5.04e3T^{2} \) |
| 73 | \( 1 + 120. iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 106. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 105. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 76.7iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 161.T + 9.40e3T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.412410812947094508502488866827, −7.66499264666693399737783768760, −6.84744947351991982992934429620, −6.18934542823014302794244597101, −5.18111528273710090722992096730, −4.43928992617329969340712680294, −3.79563436432952900835808845893, −2.75143874271377370938968581826, −1.70559821617527093180958228152, −0.36768682378295253962910745337,
1.08191344183935803125772733088, 1.84808290480501436078261382552, 3.10502886135068230218695919074, 3.80454671883565084211479334380, 4.92414370678658820278322326483, 5.59282349875147883233726856239, 6.62374662828351836294108851875, 7.07657456098047063002588807518, 7.84217920811736934998475597381, 8.548034763890415501957745073053
