| L(s) = 1 | + 2·4-s − i·7-s + 3·11-s − i·13-s + 4·16-s + 3i·17-s + 4·19-s − 9i·23-s − 2i·28-s − 6·29-s + 2·31-s − i·37-s + 3·41-s − 2i·43-s + 6·44-s + ⋯ |
| L(s) = 1 | + 4-s − 0.377i·7-s + 0.904·11-s − 0.277i·13-s + 16-s + 0.727i·17-s + 0.917·19-s − 1.87i·23-s − 0.377i·28-s − 1.11·29-s + 0.359·31-s − 0.164i·37-s + 0.468·41-s − 0.304i·43-s + 0.904·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.645692576\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.645692576\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + iT \) |
| good | 2 | \( 1 - 2T^{2} \) |
| 7 | \( 1 + iT - 7T^{2} \) |
| 11 | \( 1 - 3T + 11T^{2} \) |
| 17 | \( 1 - 3iT - 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 + 9iT - 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 + iT - 37T^{2} \) |
| 41 | \( 1 - 3T + 41T^{2} \) |
| 43 | \( 1 + 2iT - 43T^{2} \) |
| 47 | \( 1 - 6iT - 47T^{2} \) |
| 53 | \( 1 - 9iT - 53T^{2} \) |
| 59 | \( 1 + 12T + 59T^{2} \) |
| 61 | \( 1 - 5T + 61T^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 + 9T + 71T^{2} \) |
| 73 | \( 1 + 14iT - 73T^{2} \) |
| 79 | \( 1 - 7T + 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 - 15T + 89T^{2} \) |
| 97 | \( 1 - 5iT - 97T^{2} \) |
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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
−8.700736644646388665778288353120, −7.71996362240520807478426138629, −7.28975737109943529548592711289, −6.29287653522589624421246312453, −5.99476042825145240071625154211, −4.75872253365673295258341662554, −3.85275974146549053625198294568, −3.00843428348321932951962508500, −1.98410780521159862908784150188, −0.939618249599404915055090434897,
1.20423246755585730174316642969, 2.10883789920047208607630410948, 3.16709620099363009077716793628, 3.85690149488294620522114928659, 5.17040825243288585120488739694, 5.77105032823661158395318712488, 6.61376293040734706917000551051, 7.31599807008347849734486980960, 7.81449179676248811593604186449, 8.971028664256249235468784010018
