L(s) = 1 | + (−0.760 + 1.19i)2-s + (0.707 + 0.707i)3-s + (−0.844 − 1.81i)4-s + (−1.38 + 0.305i)6-s + (−0.611 + 0.611i)7-s + (2.80 + 0.371i)8-s + 1.00i·9-s + 5.12i·11-s + (0.685 − 1.87i)12-s + (−1.76 + 1.76i)13-s + (−0.264 − 1.19i)14-s + (−2.57 + 3.06i)16-s + (3.76 + 3.76i)17-s + (−1.19 − 0.760i)18-s − 1.22·19-s + ⋯ |
L(s) = 1 | + (−0.537 + 0.843i)2-s + (0.408 + 0.408i)3-s + (−0.422 − 0.906i)4-s + (−0.563 + 0.124i)6-s + (−0.231 + 0.231i)7-s + (0.991 + 0.131i)8-s + 0.333i·9-s + 1.54i·11-s + (0.197 − 0.542i)12-s + (−0.488 + 0.488i)13-s + (−0.0706 − 0.319i)14-s + (−0.643 + 0.765i)16-s + (0.912 + 0.912i)17-s + (−0.281 − 0.179i)18-s − 0.280·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.619 - 0.785i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.619 - 0.785i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.428104 + 0.882621i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.428104 + 0.882621i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.760 - 1.19i)T \) |
| 3 | \( 1 + (-0.707 - 0.707i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (0.611 - 0.611i)T - 7iT^{2} \) |
| 11 | \( 1 - 5.12iT - 11T^{2} \) |
| 13 | \( 1 + (1.76 - 1.76i)T - 13iT^{2} \) |
| 17 | \( 1 + (-3.76 - 3.76i)T + 17iT^{2} \) |
| 19 | \( 1 + 1.22T + 19T^{2} \) |
| 23 | \( 1 + (-1.07 - 1.07i)T + 23iT^{2} \) |
| 29 | \( 1 + 0.864iT - 29T^{2} \) |
| 31 | \( 1 + 7.81iT - 31T^{2} \) |
| 37 | \( 1 + (-1.76 - 1.76i)T + 37iT^{2} \) |
| 41 | \( 1 - 5.52T + 41T^{2} \) |
| 43 | \( 1 + (6.20 + 6.20i)T + 43iT^{2} \) |
| 47 | \( 1 + (2.29 - 2.29i)T - 47iT^{2} \) |
| 53 | \( 1 + (-2.62 + 2.62i)T - 53iT^{2} \) |
| 59 | \( 1 + 0.528T + 59T^{2} \) |
| 61 | \( 1 - 4.98T + 61T^{2} \) |
| 67 | \( 1 + (-6.20 + 6.20i)T - 67iT^{2} \) |
| 71 | \( 1 + 8.10iT - 71T^{2} \) |
| 73 | \( 1 + (-2.25 + 2.25i)T - 73iT^{2} \) |
| 79 | \( 1 - 15.9T + 79T^{2} \) |
| 83 | \( 1 + (-7.95 - 7.95i)T + 83iT^{2} \) |
| 89 | \( 1 + 7.25iT - 89T^{2} \) |
| 97 | \( 1 + (0.793 + 0.793i)T + 97iT^{2} \) |
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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
−12.14135191535420276293127015732, −10.76832197966024277674380336746, −9.763027637050141811950000327904, −9.434675700272127093293903100243, −8.170651096965625536654770312411, −7.40702433371978188272674840900, −6.32921943845144641275536257640, −5.10127093459004556360972450506, −4.06999611731451464439701201278, −2.04101588244523753159519780424,
0.866095057040005685096438766995, 2.74274511803492942063864279541, 3.57342429284896941660855445346, 5.23913012227238981134461460127, 6.78633712839045520103306452019, 7.86353827576345753589101572037, 8.596054214879879776513146489281, 9.556449922510788482394134511102, 10.47870184216390199949840616156, 11.37847855872475787686559494062