| L(s) = 1 | + 2·4-s − i·7-s + 11-s + i·13-s + 4·16-s + 6i·17-s + 7·19-s + 6i·23-s − 2i·28-s − 6·29-s − 7·31-s + 2i·37-s + 6·41-s + i·43-s + 2·44-s + ⋯ |
| L(s) = 1 | + 4-s − 0.377i·7-s + 0.301·11-s + 0.277i·13-s + 16-s + 1.45i·17-s + 1.60·19-s + 1.25i·23-s − 0.377i·28-s − 1.11·29-s − 1.25·31-s + 0.328i·37-s + 0.937·41-s + 0.152i·43-s + 0.301·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.464588312\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.464588312\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 - 2T^{2} \) |
| 7 | \( 1 + iT - 7T^{2} \) |
| 13 | \( 1 - iT - 13T^{2} \) |
| 17 | \( 1 - 6iT - 17T^{2} \) |
| 19 | \( 1 - 7T + 19T^{2} \) |
| 23 | \( 1 - 6iT - 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + 7T + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 - iT - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 5T + 61T^{2} \) |
| 67 | \( 1 - 5iT - 67T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 + 14iT - 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 + 6iT - 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 - 17iT - 97T^{2} \) |
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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
−9.106151442260302025402979844486, −7.931761786785549635006573289056, −7.48448661693733516914137586594, −6.76532272327669584285943252451, −5.87865026222168586660695894397, −5.30096916079741406154216323972, −3.85669466932834557595154315641, −3.40712924817309745395910884968, −2.07194957795591154620869899081, −1.25845262176396434545503229591,
0.894032717299712361175198817605, 2.21591638614725870500589481145, 2.93729826510640535921209487457, 3.88467657385001751869006516793, 5.21455883905553777613422073597, 5.67037405981371547104126721338, 6.69590727417759405042490080706, 7.32945710960031162431914299841, 7.87355519110775295375839220454, 9.034780048850486020261776817385
