| L(s) = 1 | + 2-s + 4-s − 5-s − 1.06i·7-s + 8-s − 10-s − 3.31·11-s − 3.17i·13-s − 1.06i·14-s + 16-s + 6.33i·17-s − 2.18·19-s − 20-s − 3.31·22-s + 1.65i·23-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.447·5-s − 0.401i·7-s + 0.353·8-s − 0.316·10-s − 1.00·11-s − 0.880i·13-s − 0.283i·14-s + 0.250·16-s + 1.53i·17-s − 0.500·19-s − 0.223·20-s − 0.707·22-s + 0.344i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.039911964\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.039911964\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 67 | \( 1 + (-8.09 + 1.23i)T \) |
| good | 7 | \( 1 + 1.06iT - 7T^{2} \) |
| 11 | \( 1 + 3.31T + 11T^{2} \) |
| 13 | \( 1 + 3.17iT - 13T^{2} \) |
| 17 | \( 1 - 6.33iT - 17T^{2} \) |
| 19 | \( 1 + 2.18T + 19T^{2} \) |
| 23 | \( 1 - 1.65iT - 23T^{2} \) |
| 29 | \( 1 + 10.5iT - 29T^{2} \) |
| 31 | \( 1 - 8.23iT - 31T^{2} \) |
| 37 | \( 1 - 8.87T + 37T^{2} \) |
| 41 | \( 1 + 12.0T + 41T^{2} \) |
| 43 | \( 1 - 9.77iT - 43T^{2} \) |
| 47 | \( 1 - 7.21iT - 47T^{2} \) |
| 53 | \( 1 - 9.90T + 53T^{2} \) |
| 59 | \( 1 - 5.44iT - 59T^{2} \) |
| 61 | \( 1 - 4.86iT - 61T^{2} \) |
| 71 | \( 1 + 4.71iT - 71T^{2} \) |
| 73 | \( 1 - 6.95T + 73T^{2} \) |
| 79 | \( 1 + 5.26iT - 79T^{2} \) |
| 83 | \( 1 - 10.0iT - 83T^{2} \) |
| 89 | \( 1 - 10.9iT - 89T^{2} \) |
| 97 | \( 1 - 2.27iT - 97T^{2} \) |
| 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
−8.027234791787490321491953789641, −7.62045543359574670837889012276, −6.65130610078760180573795388725, −6.00300094558205955918943919000, −5.32707235359524539210545710190, −4.48070498129129462118726377064, −3.87823469741570361692895615285, −3.05695236867441697680648237898, −2.25156548433480620342616184599, −1.00315769465277616228925581640,
0.44059612674100718735249809430, 2.02654397014431066751551247170, 2.65135463172990417826995627138, 3.55053935824589476904342164235, 4.35550967807021973751317885518, 5.14217401241662283551390667907, 5.52362482805267259019037563483, 6.70609173636222221805506462662, 7.03184585038680689894658828293, 7.84347090139256584328019039700
