L(s) = 1 | + (−0.629 − 1.26i)2-s + (−1.20 + 1.59i)4-s + 2.32i·5-s + (2.77 + 0.524i)8-s + (2.94 − 1.46i)10-s − 3.58i·11-s − 2.93i·13-s + (−1.08 − 3.84i)16-s + 2.32i·17-s − 8.33·19-s + (−3.71 − 2.80i)20-s + (−4.53 + 2.25i)22-s − 1.48i·23-s − 0.414·25-s + (−3.71 + 1.84i)26-s + ⋯ |
L(s) = 1 | + (−0.445 − 0.895i)2-s + (−0.603 + 0.797i)4-s + 1.04i·5-s + (0.982 + 0.185i)8-s + (0.931 − 0.463i)10-s − 1.07i·11-s − 0.812i·13-s + (−0.271 − 0.962i)16-s + 0.564i·17-s − 1.91·19-s + (−0.829 − 0.628i)20-s + (−0.966 + 0.480i)22-s − 0.309i·23-s − 0.0828·25-s + (−0.727 + 0.361i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.224 + 0.974i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.224 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9649550075\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9649550075\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.629 + 1.26i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 - 2.32iT - 5T^{2} \) |
| 11 | \( 1 + 3.58iT - 11T^{2} \) |
| 13 | \( 1 + 2.93iT - 13T^{2} \) |
| 17 | \( 1 - 2.32iT - 17T^{2} \) |
| 19 | \( 1 + 8.33T + 19T^{2} \) |
| 23 | \( 1 + 1.48iT - 23T^{2} \) |
| 29 | \( 1 - 7.86T + 29T^{2} \) |
| 31 | \( 1 - 3.45T + 31T^{2} \) |
| 37 | \( 1 + 8.24T + 37T^{2} \) |
| 41 | \( 1 + 2.89iT - 41T^{2} \) |
| 43 | \( 1 + 6.37iT - 43T^{2} \) |
| 47 | \( 1 - 10.4T + 47T^{2} \) |
| 53 | \( 1 - 8.59T + 53T^{2} \) |
| 59 | \( 1 + 10.4T + 59T^{2} \) |
| 61 | \( 1 - 2.93iT - 61T^{2} \) |
| 67 | \( 1 + 9.02iT - 67T^{2} \) |
| 71 | \( 1 + 10.7iT - 71T^{2} \) |
| 73 | \( 1 + 11.2iT - 73T^{2} \) |
| 79 | \( 1 + 15.3iT - 79T^{2} \) |
| 83 | \( 1 - 14.8T + 83T^{2} \) |
| 89 | \( 1 - 13.5iT - 89T^{2} \) |
| 97 | \( 1 + 5.35iT - 97T^{2} \) |
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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
−8.915102729936194597775815344463, −8.530233336843525050777465024752, −7.71521424776271505955317973955, −6.70295550891889107743323734547, −5.98894709251861157120270622766, −4.72030363450619593979851421132, −3.68393276887592424095781027015, −2.97721102889748389487957923423, −2.09751363874314606187620955883, −0.48959446679038249125839966977,
1.10363390315352232626702084212, 2.25587598349830082165641903102, 4.27899149507433422377944966363, 4.57449924350644325663591301218, 5.48103120428632071522672456689, 6.57109057056548299159471732682, 7.02379918230400017840373342756, 8.115385599629853228425016420418, 8.681404926187855112876235601458, 9.275162270207830496178264329792