| L(s) = 1 | + (−0.655 + 2.13i)5-s + (−2.09 + 1.61i)7-s − 6.16i·11-s + 0.742·13-s + 3.80i·17-s − 7.08i·19-s + 5.47·23-s + (−4.13 − 2.80i)25-s + 3.48i·29-s + 7.72i·31-s + (−2.08 − 5.53i)35-s − 4.20i·37-s − 1.52·41-s − 12.0i·43-s − 0.165i·47-s + ⋯ |
| L(s) = 1 | + (−0.293 + 0.956i)5-s + (−0.791 + 0.611i)7-s − 1.85i·11-s + 0.205·13-s + 0.924i·17-s − 1.62i·19-s + 1.14·23-s + (−0.827 − 0.560i)25-s + 0.646i·29-s + 1.38i·31-s + (−0.352 − 0.935i)35-s − 0.691i·37-s − 0.238·41-s − 1.83i·43-s − 0.0240i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 + 0.252i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.967 + 0.252i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.352931302\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.352931302\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (0.655 - 2.13i)T \) |
| 7 | \( 1 + (2.09 - 1.61i)T \) |
| good | 11 | \( 1 + 6.16iT - 11T^{2} \) |
| 13 | \( 1 - 0.742T + 13T^{2} \) |
| 17 | \( 1 - 3.80iT - 17T^{2} \) |
| 19 | \( 1 + 7.08iT - 19T^{2} \) |
| 23 | \( 1 - 5.47T + 23T^{2} \) |
| 29 | \( 1 - 3.48iT - 29T^{2} \) |
| 31 | \( 1 - 7.72iT - 31T^{2} \) |
| 37 | \( 1 + 4.20iT - 37T^{2} \) |
| 41 | \( 1 + 1.52T + 41T^{2} \) |
| 43 | \( 1 + 12.0iT - 43T^{2} \) |
| 47 | \( 1 + 0.165iT - 47T^{2} \) |
| 53 | \( 1 + 1.52T + 53T^{2} \) |
| 59 | \( 1 - 11.2T + 59T^{2} \) |
| 61 | \( 1 + 7.03iT - 61T^{2} \) |
| 67 | \( 1 - 0.383iT - 67T^{2} \) |
| 71 | \( 1 - 5.81iT - 71T^{2} \) |
| 73 | \( 1 - 11.8T + 73T^{2} \) |
| 79 | \( 1 - 10.5T + 79T^{2} \) |
| 83 | \( 1 - 10.5iT - 83T^{2} \) |
| 89 | \( 1 - 0.779T + 89T^{2} \) |
| 97 | \( 1 + 7.92T + 97T^{2} \) |
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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
−8.700964152001864770483124575702, −8.413469403440774913799256673910, −7.04219008954567099372363956777, −6.70437587678396588248130735036, −5.85317665635226472042285997045, −5.13924633129033272765591621987, −3.62316986616763692990794536184, −3.26840296194857012599390156913, −2.37353860754055628340366058178, −0.59885605849653268892574278983,
0.896202762609678774838753798585, 2.05734822212614617610432848624, 3.35693197263476101838506849309, 4.29436429874421447401563953896, 4.78426388804077816790869830492, 5.82743040062470697943296320150, 6.73716491616897986194394640121, 7.52159523267010480466376615946, 8.000784232050974117454159112979, 9.110928377693416552749836395474
