| L(s) = 1 | − 2i·5-s + 2i·7-s + (−3 − 2i)13-s + 2·17-s + 6i·19-s + 4·23-s + 25-s + 10·29-s − 10i·31-s + 4·35-s + 8i·37-s − 10i·41-s − 4·43-s − 12i·47-s + 3·49-s + ⋯ |
| L(s) = 1 | − 0.894i·5-s + 0.755i·7-s + (−0.832 − 0.554i)13-s + 0.485·17-s + 1.37i·19-s + 0.834·23-s + 0.200·25-s + 1.85·29-s − 1.79i·31-s + 0.676·35-s + 1.31i·37-s − 1.56i·41-s − 0.609·43-s − 1.75i·47-s + 0.428·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 + 0.554i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.832 + 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.688611181\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.688611181\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 + (3 + 2i)T \) |
| good | 5 | \( 1 + 2iT - 5T^{2} \) |
| 7 | \( 1 - 2iT - 7T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 - 6iT - 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 - 10T + 29T^{2} \) |
| 31 | \( 1 + 10iT - 31T^{2} \) |
| 37 | \( 1 - 8iT - 37T^{2} \) |
| 41 | \( 1 + 10iT - 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 + 12iT - 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 - 4iT - 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 - 2iT - 67T^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + 4iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 4iT - 83T^{2} \) |
| 89 | \( 1 - 6iT - 89T^{2} \) |
| 97 | \( 1 + 12iT - 97T^{2} \) |
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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
−9.007385919229748411351164434573, −8.447263688715366358683879441083, −7.77692294713283931596655897320, −6.78018313711248701755063961250, −5.70819126897290076721028271593, −5.22682445916237200712828707122, −4.33172448019589934184046759446, −3.16537262543540868017865575838, −2.13198175037760075101119464600, −0.794807746838400400493292765771,
1.02058498317847053628252469040, 2.59868470270459072400301490415, 3.21863537722915619675050225897, 4.48232571967047506210393698411, 5.06156645094311968166905732233, 6.45375414631718386441229477036, 6.93719216554941773200738777003, 7.48497016770577177653000297543, 8.553501971817179384632095504635, 9.361995907518608237787438983653
