| L(s) = 1 | + 2.64i·2-s + i·3-s − 5.00·4-s − 2.64·6-s − 3.64i·7-s − 7.93i·8-s − 9-s + 5.64·11-s − 5.00i·12-s + 5.64i·13-s + 9.64·14-s + 11.0·16-s − 4i·17-s − 2.64i·18-s + 19-s + ⋯ |
| L(s) = 1 | + 1.87i·2-s + 0.577i·3-s − 2.50·4-s − 1.08·6-s − 1.37i·7-s − 2.80i·8-s − 0.333·9-s + 1.70·11-s − 1.44i·12-s + 1.56i·13-s + 2.57·14-s + 2.75·16-s − 0.970i·17-s − 0.623i·18-s + 0.229·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.515545968\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.515545968\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 - 2.64iT - 2T^{2} \) |
| 7 | \( 1 + 3.64iT - 7T^{2} \) |
| 11 | \( 1 - 5.64T + 11T^{2} \) |
| 13 | \( 1 - 5.64iT - 13T^{2} \) |
| 17 | \( 1 + 4iT - 17T^{2} \) |
| 23 | \( 1 - 1.29iT - 23T^{2} \) |
| 29 | \( 1 - 6.93T + 29T^{2} \) |
| 31 | \( 1 - 6T + 31T^{2} \) |
| 37 | \( 1 + 1.64iT - 37T^{2} \) |
| 41 | \( 1 + 4.35T + 41T^{2} \) |
| 43 | \( 1 + 0.354iT - 43T^{2} \) |
| 47 | \( 1 - 9.29iT - 47T^{2} \) |
| 53 | \( 1 + 0.708iT - 53T^{2} \) |
| 59 | \( 1 + 0.708T + 59T^{2} \) |
| 61 | \( 1 + 0.708T + 61T^{2} \) |
| 67 | \( 1 - 14.5iT - 67T^{2} \) |
| 71 | \( 1 + 3.29T + 71T^{2} \) |
| 73 | \( 1 + 10iT - 73T^{2} \) |
| 79 | \( 1 - 14.5T + 79T^{2} \) |
| 83 | \( 1 + 6iT - 83T^{2} \) |
| 89 | \( 1 - 1.06T + 89T^{2} \) |
| 97 | \( 1 - 12.9iT - 97T^{2} \) |
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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
−9.450271011683333976194610807093, −9.084262970852396108513543404233, −8.153003260342570139826533489236, −7.14085292957193922158697181859, −6.77306864104276983433946453203, −6.08465483102187199736904327314, −4.67173632818568316080806737483, −4.42609621640050870761065964335, −3.56865178314112894591400262752, −1.00592871596478144731837563201,
0.884996695984924135403672373460, 1.89395016087929708957627954000, 2.85540761259727915283466745800, 3.58859458748282625006603076806, 4.76076014315921208722936644579, 5.70881982029232643084603607904, 6.54170038930594026168364181781, 8.256913737782010434068908369797, 8.495845679306433854419339447038, 9.360143914069802781754932282640
