| L(s) = 1 | + (0.734 + 0.734i)3-s − 1.71i·7-s − 1.92i·9-s + (−2.82 + 2.82i)11-s + (2.59 + 2.59i)13-s − 1.89·17-s + (2.89 + 2.89i)19-s + (1.25 − 1.25i)21-s + 2.00i·23-s + (3.61 − 3.61i)27-s + (6.72 + 6.72i)29-s + 7.11·31-s − 4.15·33-s + (−2.25 + 2.25i)37-s + 3.81i·39-s + ⋯ |
| L(s) = 1 | + (0.423 + 0.423i)3-s − 0.648i·7-s − 0.640i·9-s + (−0.852 + 0.852i)11-s + (0.719 + 0.719i)13-s − 0.460·17-s + (0.664 + 0.664i)19-s + (0.274 − 0.274i)21-s + 0.418i·23-s + (0.695 − 0.695i)27-s + (1.24 + 1.24i)29-s + 1.27·31-s − 0.722·33-s + (−0.370 + 0.370i)37-s + 0.610i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.813 - 0.581i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.813 - 0.581i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.956381343\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.956381343\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + (-0.734 - 0.734i)T + 3iT^{2} \) |
| 7 | \( 1 + 1.71iT - 7T^{2} \) |
| 11 | \( 1 + (2.82 - 2.82i)T - 11iT^{2} \) |
| 13 | \( 1 + (-2.59 - 2.59i)T + 13iT^{2} \) |
| 17 | \( 1 + 1.89T + 17T^{2} \) |
| 19 | \( 1 + (-2.89 - 2.89i)T + 19iT^{2} \) |
| 23 | \( 1 - 2.00iT - 23T^{2} \) |
| 29 | \( 1 + (-6.72 - 6.72i)T + 29iT^{2} \) |
| 31 | \( 1 - 7.11T + 31T^{2} \) |
| 37 | \( 1 + (2.25 - 2.25i)T - 37iT^{2} \) |
| 41 | \( 1 - 1.59iT - 41T^{2} \) |
| 43 | \( 1 + (-8.06 + 8.06i)T - 43iT^{2} \) |
| 47 | \( 1 - 4.43T + 47T^{2} \) |
| 53 | \( 1 + (-0.481 + 0.481i)T - 53iT^{2} \) |
| 59 | \( 1 + (-3.08 + 3.08i)T - 59iT^{2} \) |
| 61 | \( 1 + (-3.46 - 3.46i)T + 61iT^{2} \) |
| 67 | \( 1 + (1.80 + 1.80i)T + 67iT^{2} \) |
| 71 | \( 1 + 0.379iT - 71T^{2} \) |
| 73 | \( 1 + 8.37iT - 73T^{2} \) |
| 79 | \( 1 - 11.2T + 79T^{2} \) |
| 83 | \( 1 + (8.24 + 8.24i)T + 83iT^{2} \) |
| 89 | \( 1 - 11.9iT - 89T^{2} \) |
| 97 | \( 1 + 6.50T + 97T^{2} \) |
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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
−9.461311785443866146705559725213, −8.790714986175038777318058976405, −7.959353630251339112985485883570, −7.06591933236669514881646369759, −6.41182751780715567582986878594, −5.23049335364455821484348627492, −4.32007622244572065892027861505, −3.62514985426174556566635915982, −2.56257689333021650784639519677, −1.14822872930529835725805560578,
0.874827671805461285769377738999, 2.54004639556202032391441583225, 2.84993733285397654649117940910, 4.32869105150017937558529577660, 5.35196012246332383474968575958, 5.99687092486948259651164590851, 6.99616093856647777774494824027, 8.088918040296082169251192946723, 8.267185871466120249737076860998, 9.122606514826480926184318520030
