L(s) = 1 | + (0.817 − 2.08i)5-s + 4.41i·7-s − 2.29·11-s − 6.92i·13-s − 1.51i·17-s − 2.89·19-s + i·23-s + (−3.66 − 3.40i)25-s − 7.68·29-s + 3.85·31-s + (9.18 + 3.60i)35-s + 8.62i·37-s + 6.44·41-s + 3.48i·43-s + 6.19i·47-s + ⋯ |
L(s) = 1 | + (0.365 − 0.930i)5-s + 1.66i·7-s − 0.691·11-s − 1.92i·13-s − 0.367i·17-s − 0.665·19-s + 0.208i·23-s + (−0.732 − 0.680i)25-s − 1.42·29-s + 0.692·31-s + (1.55 + 0.609i)35-s + 1.41i·37-s + 1.00·41-s + 0.531i·43-s + 0.903i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.930 - 0.365i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.930 - 0.365i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1408632663\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1408632663\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.817 + 2.08i)T \) |
| 23 | \( 1 - iT \) |
good | 7 | \( 1 - 4.41iT - 7T^{2} \) |
| 11 | \( 1 + 2.29T + 11T^{2} \) |
| 13 | \( 1 + 6.92iT - 13T^{2} \) |
| 17 | \( 1 + 1.51iT - 17T^{2} \) |
| 19 | \( 1 + 2.89T + 19T^{2} \) |
| 29 | \( 1 + 7.68T + 29T^{2} \) |
| 31 | \( 1 - 3.85T + 31T^{2} \) |
| 37 | \( 1 - 8.62iT - 37T^{2} \) |
| 41 | \( 1 - 6.44T + 41T^{2} \) |
| 43 | \( 1 - 3.48iT - 43T^{2} \) |
| 47 | \( 1 - 6.19iT - 47T^{2} \) |
| 53 | \( 1 - 2.17iT - 53T^{2} \) |
| 59 | \( 1 + 11.7T + 59T^{2} \) |
| 61 | \( 1 + 5.11T + 61T^{2} \) |
| 67 | \( 1 - 9.94iT - 67T^{2} \) |
| 71 | \( 1 + 3.41T + 71T^{2} \) |
| 73 | \( 1 - 8.95iT - 73T^{2} \) |
| 79 | \( 1 - 1.92T + 79T^{2} \) |
| 83 | \( 1 - 8.04iT - 83T^{2} \) |
| 89 | \( 1 + 1.09T + 89T^{2} \) |
| 97 | \( 1 + 16.9iT - 97T^{2} \) |
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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
−8.600406235056080374567764170936, −8.194855266025673353664060159691, −7.54666560892448083394600224775, −6.15876754570665708209286330094, −5.70647897518453060074355485386, −5.22069480854226531537058276008, −4.43648592896756318925616782765, −3.00065309657832481704871899596, −2.55522435632986980530060015792, −1.37556843612408292970145288129,
0.03752494361453562399073407931, 1.63567043620041701107223041886, 2.39802684251788955132224781395, 3.69843010454443344190474566200, 4.07046847086904874880016375730, 4.99599644964996665057914433844, 6.21207032618176682312661645246, 6.59682937046933942735630838833, 7.46398064782765396317679039458, 7.69499241258641379340650570775