| L(s) = 1 | + 2.60i·2-s + (−2.93 − 0.635i)3-s − 2.80·4-s + (1.65 − 7.65i)6-s + 2.64·7-s + 3.10i·8-s + (8.19 + 3.72i)9-s − 7.66i·11-s + (8.23 + 1.78i)12-s − 12.2·13-s + 6.90i·14-s − 19.3·16-s − 28.0i·17-s + (−9.72 + 21.3i)18-s − 22.7·19-s + ⋯ |
| L(s) = 1 | + 1.30i·2-s + (−0.977 − 0.211i)3-s − 0.702·4-s + (0.276 − 1.27i)6-s + 0.377·7-s + 0.388i·8-s + (0.910 + 0.414i)9-s − 0.697i·11-s + (0.686 + 0.148i)12-s − 0.944·13-s + 0.493i·14-s − 1.20·16-s − 1.64i·17-s + (−0.540 + 1.18i)18-s − 1.19·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 + 0.211i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.977 + 0.211i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.8317468552\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8317468552\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (2.93 + 0.635i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - 2.64T \) |
| good | 2 | \( 1 - 2.60iT - 4T^{2} \) |
| 11 | \( 1 + 7.66iT - 121T^{2} \) |
| 13 | \( 1 + 12.2T + 169T^{2} \) |
| 17 | \( 1 + 28.0iT - 289T^{2} \) |
| 19 | \( 1 + 22.7T + 361T^{2} \) |
| 23 | \( 1 + 18.7iT - 529T^{2} \) |
| 29 | \( 1 - 13.5iT - 841T^{2} \) |
| 31 | \( 1 - 59.4T + 961T^{2} \) |
| 37 | \( 1 - 28.7T + 1.36e3T^{2} \) |
| 41 | \( 1 + 11.5iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 4.45T + 1.84e3T^{2} \) |
| 47 | \( 1 + 6.20iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 38.3iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 106. iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 55.5T + 3.72e3T^{2} \) |
| 67 | \( 1 - 87.8T + 4.48e3T^{2} \) |
| 71 | \( 1 + 29.0iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 0.585T + 5.32e3T^{2} \) |
| 79 | \( 1 - 7.71T + 6.24e3T^{2} \) |
| 83 | \( 1 + 62.0iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 150. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 36.3T + 9.40e3T^{2} \) |
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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
−10.74029883582034920796419848836, −9.651648888391392014664190458901, −8.484125932024373526454898841663, −7.69146551325435275153765866434, −6.79986474139516016140212546312, −6.21941651118341445330006045236, −5.09837011655276234910547193178, −4.61002123558271076146434522242, −2.43799545946145110506481279265, −0.39449702715609178008467447782,
1.28310473248576391175346870310, 2.41697620528274109417386668278, 4.05545551105117962912720613751, 4.60175574361905105895651090441, 5.97476424950401944219000159469, 6.89799372597404585005064372612, 8.086132961119714824742505192074, 9.452868684089649190995157609895, 10.14932602911623305152373530386, 10.70158686111399008371356110191
