| L(s) = 1 | + (−1.32 + 1.11i)3-s − 2.23i·5-s − 2.64·7-s + (0.5 − 2.95i)9-s − 5.91i·11-s − 2.64·13-s + (2.50 + 2.95i)15-s + 2.23i·17-s + (3.50 − 2.95i)21-s − 5.00·25-s + (2.64 + 4.47i)27-s + 5.91i·29-s + (6.61 + 7.82i)33-s + 5.91i·35-s + (3.50 − 2.95i)39-s + ⋯ |
| L(s) = 1 | + (−0.763 + 0.645i)3-s − 0.999i·5-s − 0.999·7-s + (0.166 − 0.986i)9-s − 1.78i·11-s − 0.733·13-s + (0.645 + 0.763i)15-s + 0.542i·17-s + (0.763 − 0.645i)21-s − 1.00·25-s + (0.509 + 0.860i)27-s + 1.09i·29-s + (1.15 + 1.36i)33-s + 0.999i·35-s + (0.560 − 0.473i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.645 - 0.763i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.645 - 0.763i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1616255540\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1616255540\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.32 - 1.11i)T \) |
| 5 | \( 1 + 2.23iT \) |
| 7 | \( 1 + 2.64T \) |
| good | 11 | \( 1 + 5.91iT - 11T^{2} \) |
| 13 | \( 1 + 2.64T + 13T^{2} \) |
| 17 | \( 1 - 2.23iT - 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 5.91iT - 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 - 11.1iT - 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 - 11.8iT - 71T^{2} \) |
| 73 | \( 1 + 10.5T + 73T^{2} \) |
| 79 | \( 1 - T + 79T^{2} \) |
| 83 | \( 1 + 8.94iT - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 18.5T + 97T^{2} \) |
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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
−9.590194101128281542752228077783, −8.975812006353937238102312862966, −8.317861108122142254333921278450, −7.11094526708517149361468758333, −6.07152747577467004147520299713, −5.69808629692375829072147900302, −4.75683133403135700089606486296, −3.82129032437384323329234736002, −3.00067098886229579902133289545, −1.06095138631415607151277542737,
0.07773403085298341855658311822, 2.00229986031107690271895440234, 2.73555401149597140367336528649, 4.07953120959303464255623529220, 5.05159281374872338844494724729, 6.02149240353965014746300686090, 6.86236835312413898664527206301, 7.15708660608576120217913140246, 7.895082959824412425377174337841, 9.367308150241458349083125084667
