L(s) = 1 | − 1.86i·3-s + 2.89i·5-s + 5.10i·7-s − 0.459·9-s + 3.69i·11-s + 2.99·13-s + 5.37·15-s + (1.11 − 3.96i)17-s + 6.95·19-s + 9.50·21-s + 5.32i·23-s − 3.35·25-s − 4.72i·27-s + 2.80i·29-s + 1.47i·31-s + ⋯ |
L(s) = 1 | − 1.07i·3-s + 1.29i·5-s + 1.93i·7-s − 0.153·9-s + 1.11i·11-s + 0.831·13-s + 1.38·15-s + (0.271 − 0.962i)17-s + 1.59·19-s + 2.07·21-s + 1.10i·23-s − 0.671·25-s − 0.909i·27-s + 0.520i·29-s + 0.264i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.271 - 0.962i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.271 - 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.217078774\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.217078774\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 17 | \( 1 + (-1.11 + 3.96i)T \) |
| 59 | \( 1 + T \) |
good | 3 | \( 1 + 1.86iT - 3T^{2} \) |
| 5 | \( 1 - 2.89iT - 5T^{2} \) |
| 7 | \( 1 - 5.10iT - 7T^{2} \) |
| 11 | \( 1 - 3.69iT - 11T^{2} \) |
| 13 | \( 1 - 2.99T + 13T^{2} \) |
| 19 | \( 1 - 6.95T + 19T^{2} \) |
| 23 | \( 1 - 5.32iT - 23T^{2} \) |
| 29 | \( 1 - 2.80iT - 29T^{2} \) |
| 31 | \( 1 - 1.47iT - 31T^{2} \) |
| 37 | \( 1 + 0.220iT - 37T^{2} \) |
| 41 | \( 1 + 10.9iT - 41T^{2} \) |
| 43 | \( 1 - 5.46T + 43T^{2} \) |
| 47 | \( 1 - 2.85T + 47T^{2} \) |
| 53 | \( 1 - 13.5T + 53T^{2} \) |
| 61 | \( 1 - 6.60iT - 61T^{2} \) |
| 67 | \( 1 - 8.19T + 67T^{2} \) |
| 71 | \( 1 + 7.53iT - 71T^{2} \) |
| 73 | \( 1 - 12.2iT - 73T^{2} \) |
| 79 | \( 1 + 0.707iT - 79T^{2} \) |
| 83 | \( 1 + 2.96T + 83T^{2} \) |
| 89 | \( 1 - 4.12T + 89T^{2} \) |
| 97 | \( 1 + 0.479iT - 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.546862311892403552120552676756, −7.50330776627124599469357979626, −7.23651338022940175216075284845, −6.56088323251991881533415030055, −5.63789092342073044397983552704, −5.28359563610570126487585894822, −3.75179592627303358215528813042, −2.78408196143070461833316428544, −2.31306467892962592693651880242, −1.33071886304371482600449721479,
0.77741625366717439093712956418, 1.23009021712076956449945009195, 3.22066467778030906902917216222, 3.93567172452401398037787199963, 4.31071382804138807740964992726, 5.11854131157529246264839779338, 5.90150222344355265550343048839, 6.82335038904633144565737424992, 7.83533231564165968620884080232, 8.290678724931089539290228508096