L(s) = 1 | + (1.39 + 0.215i)2-s + (1.90 + 0.602i)4-s + (1.33 + 1.33i)5-s + 0.400·7-s + (2.53 + 1.25i)8-s + (1.58 + 2.15i)10-s + (−0.0888 + 0.0888i)11-s + (−3.59 − 3.59i)13-s + (0.559 + 0.0863i)14-s + (3.27 + 2.29i)16-s − 0.898i·17-s + (−3.16 + 3.16i)19-s + (1.74 + 3.35i)20-s + (−0.143 + 0.105i)22-s + 7.09i·23-s + ⋯ |
L(s) = 1 | + (0.988 + 0.152i)2-s + (0.953 + 0.301i)4-s + (0.598 + 0.598i)5-s + 0.151·7-s + (0.896 + 0.443i)8-s + (0.500 + 0.682i)10-s + (−0.0267 + 0.0267i)11-s + (−0.998 − 0.998i)13-s + (0.149 + 0.0230i)14-s + (0.818 + 0.574i)16-s − 0.217i·17-s + (−0.726 + 0.726i)19-s + (0.390 + 0.750i)20-s + (−0.0305 + 0.0223i)22-s + 1.47i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.844 - 0.536i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.844 - 0.536i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.60874 + 0.758664i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.60874 + 0.758664i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.39 - 0.215i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-1.33 - 1.33i)T + 5iT^{2} \) |
| 7 | \( 1 - 0.400T + 7T^{2} \) |
| 11 | \( 1 + (0.0888 - 0.0888i)T - 11iT^{2} \) |
| 13 | \( 1 + (3.59 + 3.59i)T + 13iT^{2} \) |
| 17 | \( 1 + 0.898iT - 17T^{2} \) |
| 19 | \( 1 + (3.16 - 3.16i)T - 19iT^{2} \) |
| 23 | \( 1 - 7.09iT - 23T^{2} \) |
| 29 | \( 1 + (-5.95 + 5.95i)T - 29iT^{2} \) |
| 31 | \( 1 + 5.25iT - 31T^{2} \) |
| 37 | \( 1 + (-0.934 + 0.934i)T - 37iT^{2} \) |
| 41 | \( 1 - 5.47T + 41T^{2} \) |
| 43 | \( 1 + (6.81 + 6.81i)T + 43iT^{2} \) |
| 47 | \( 1 + 6.60T + 47T^{2} \) |
| 53 | \( 1 + (-3.33 - 3.33i)T + 53iT^{2} \) |
| 59 | \( 1 + (4.12 - 4.12i)T - 59iT^{2} \) |
| 61 | \( 1 + (7.11 + 7.11i)T + 61iT^{2} \) |
| 67 | \( 1 + (1.42 - 1.42i)T - 67iT^{2} \) |
| 71 | \( 1 + 0.567iT - 71T^{2} \) |
| 73 | \( 1 - 12.1iT - 73T^{2} \) |
| 79 | \( 1 + 4.45iT - 79T^{2} \) |
| 83 | \( 1 + (6.47 + 6.47i)T + 83iT^{2} \) |
| 89 | \( 1 + 6.04T + 89T^{2} \) |
| 97 | \( 1 - 15.8T + 97T^{2} \) |
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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
−11.39796511883156337335496287341, −10.36049464153221931155502901051, −9.802232302900341395189925818831, −8.141033918203050740631984930156, −7.40838770574420568952752280037, −6.30087097138050514004306981355, −5.57820365490801458552992448609, −4.47379463781167013249199621354, −3.14588310813891749134053757343, −2.12127054176080006774140680577,
1.67519930426694952005988185240, 2.88753933770082013880404182325, 4.56527259780842976359262822896, 4.94676401617772231029293654951, 6.32303371592058685720352834445, 6.97086225473854969204490630587, 8.361264718920180673733318976739, 9.376161380452736610001580003192, 10.34576922324444196563264697793, 11.19572267082799815507205659893