L(s) = 1 | + (−0.707 − 0.707i)3-s + (−0.770 + 2.09i)5-s + 3.05·7-s + 1.00i·9-s + (−1.80 − 1.80i)11-s + (2.47 + 2.47i)13-s + (2.02 − 0.939i)15-s + 3.66i·17-s + (−2.31 + 2.31i)19-s + (−2.15 − 2.15i)21-s + 4.86·23-s + (−3.81 − 3.23i)25-s + (0.707 − 0.707i)27-s + (−4.74 + 4.74i)29-s − 1.86·31-s + ⋯ |
L(s) = 1 | + (−0.408 − 0.408i)3-s + (−0.344 + 0.938i)5-s + 1.15·7-s + 0.333i·9-s + (−0.545 − 0.545i)11-s + (0.685 + 0.685i)13-s + (0.523 − 0.242i)15-s + 0.889i·17-s + (−0.531 + 0.531i)19-s + (−0.470 − 0.470i)21-s + 1.01·23-s + (−0.762 − 0.646i)25-s + (0.136 − 0.136i)27-s + (−0.881 + 0.881i)29-s − 0.334·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0515 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0515 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.281340960\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.281340960\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.707 + 0.707i)T \) |
| 5 | \( 1 + (0.770 - 2.09i)T \) |
good | 7 | \( 1 - 3.05T + 7T^{2} \) |
| 11 | \( 1 + (1.80 + 1.80i)T + 11iT^{2} \) |
| 13 | \( 1 + (-2.47 - 2.47i)T + 13iT^{2} \) |
| 17 | \( 1 - 3.66iT - 17T^{2} \) |
| 19 | \( 1 + (2.31 - 2.31i)T - 19iT^{2} \) |
| 23 | \( 1 - 4.86T + 23T^{2} \) |
| 29 | \( 1 + (4.74 - 4.74i)T - 29iT^{2} \) |
| 31 | \( 1 + 1.86T + 31T^{2} \) |
| 37 | \( 1 + (-5.40 + 5.40i)T - 37iT^{2} \) |
| 41 | \( 1 + 6.47iT - 41T^{2} \) |
| 43 | \( 1 + (-4.19 + 4.19i)T - 43iT^{2} \) |
| 47 | \( 1 - 8.24iT - 47T^{2} \) |
| 53 | \( 1 + (9.99 - 9.99i)T - 53iT^{2} \) |
| 59 | \( 1 + (2.47 + 2.47i)T + 59iT^{2} \) |
| 61 | \( 1 + (8.01 - 8.01i)T - 61iT^{2} \) |
| 67 | \( 1 + (-8.60 - 8.60i)T + 67iT^{2} \) |
| 71 | \( 1 - 6.63iT - 71T^{2} \) |
| 73 | \( 1 + 2.70T + 73T^{2} \) |
| 79 | \( 1 - 10.9T + 79T^{2} \) |
| 83 | \( 1 + (3.65 + 3.65i)T + 83iT^{2} \) |
| 89 | \( 1 - 13.1iT - 89T^{2} \) |
| 97 | \( 1 - 12.4iT - 97T^{2} \) |
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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
−9.256002659574679201236468478713, −8.437079651666752989364506325891, −7.74401050065638716814399491905, −7.13595754239430980209943789650, −6.16508902075035862451584636599, −5.58402457655076743276088700546, −4.43801626121778166627110627978, −3.60692112035546965234339397661, −2.37543142956688265255035053107, −1.35684940782660395481006236043,
0.52374352728792940605776382663, 1.76301630756168241235936724426, 3.16863616074711882263604379310, 4.42532980696095273465340685382, 4.87164799033008201971095972150, 5.49271326286406034775687183943, 6.59144844971001654930401167061, 7.81430414018671620512797777146, 8.043509734131811685015098231464, 9.086712148068948619591113923847