| L(s) = 1 | + (1.39 + 0.230i)2-s + (−0.794 + 2.40i)3-s + (1.89 + 0.643i)4-s + (−1.03 + 2.68i)5-s + (−1.66 + 3.17i)6-s + (0.286 − 0.477i)7-s + (2.49 + 1.33i)8-s + (−2.74 − 2.03i)9-s + (−2.06 + 3.50i)10-s + (−0.119 − 0.970i)11-s + (−3.05 + 4.04i)12-s + (0.00544 − 0.221i)13-s + (0.509 − 0.600i)14-s + (−5.62 − 4.61i)15-s + (3.17 + 2.43i)16-s + (−1.14 − 1.40i)17-s + ⋯ |
| L(s) = 1 | + (0.986 + 0.163i)2-s + (−0.458 + 1.38i)3-s + (0.946 + 0.321i)4-s + (−0.462 + 1.19i)5-s + (−0.678 + 1.29i)6-s + (0.108 − 0.180i)7-s + (0.881 + 0.471i)8-s + (−0.913 − 0.677i)9-s + (−0.652 + 1.10i)10-s + (−0.0360 − 0.292i)11-s + (−0.881 + 1.16i)12-s + (0.00151 − 0.0615i)13-s + (0.136 − 0.160i)14-s + (−1.45 − 1.19i)15-s + (0.792 + 0.609i)16-s + (−0.278 − 0.339i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.763 - 0.645i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.763 - 0.645i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.717018 + 1.95980i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.717018 + 1.95980i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.39 - 0.230i)T \) |
| good | 3 | \( 1 + (0.794 - 2.40i)T + (-2.40 - 1.78i)T^{2} \) |
| 5 | \( 1 + (1.03 - 2.68i)T + (-3.70 - 3.35i)T^{2} \) |
| 7 | \( 1 + (-0.286 + 0.477i)T + (-3.29 - 6.17i)T^{2} \) |
| 11 | \( 1 + (0.119 + 0.970i)T + (-10.6 + 2.67i)T^{2} \) |
| 13 | \( 1 + (-0.00544 + 0.221i)T + (-12.9 - 0.637i)T^{2} \) |
| 17 | \( 1 + (1.14 + 1.40i)T + (-3.31 + 16.6i)T^{2} \) |
| 19 | \( 1 + (-1.31 - 0.926i)T + (6.40 + 17.8i)T^{2} \) |
| 23 | \( 1 + (-2.73 + 5.78i)T + (-14.5 - 17.7i)T^{2} \) |
| 29 | \( 1 + (2.68 - 0.741i)T + (24.8 - 14.9i)T^{2} \) |
| 31 | \( 1 + (-1.16 + 0.231i)T + (28.6 - 11.8i)T^{2} \) |
| 37 | \( 1 + (-1.43 - 6.39i)T + (-33.4 + 15.8i)T^{2} \) |
| 41 | \( 1 + (2.61 - 2.36i)T + (4.01 - 40.8i)T^{2} \) |
| 43 | \( 1 + (-1.90 + 3.78i)T + (-25.6 - 34.5i)T^{2} \) |
| 47 | \( 1 + (4.52 + 2.41i)T + (26.1 + 39.0i)T^{2} \) |
| 53 | \( 1 + (-0.774 - 0.214i)T + (45.4 + 27.2i)T^{2} \) |
| 59 | \( 1 + (-11.9 + 0.292i)T + (58.9 - 2.89i)T^{2} \) |
| 61 | \( 1 + (-6.92 - 5.97i)T + (8.95 + 60.3i)T^{2} \) |
| 67 | \( 1 + (-4.62 + 0.340i)T + (66.2 - 9.83i)T^{2} \) |
| 71 | \( 1 + (-1.99 + 13.4i)T + (-67.9 - 20.6i)T^{2} \) |
| 73 | \( 1 + (-6.35 - 10.6i)T + (-34.4 + 64.3i)T^{2} \) |
| 79 | \( 1 + (15.6 + 4.74i)T + (65.6 + 43.8i)T^{2} \) |
| 83 | \( 1 + (0.0662 - 0.295i)T + (-75.0 - 35.4i)T^{2} \) |
| 89 | \( 1 + (1.87 + 3.96i)T + (-56.4 + 68.7i)T^{2} \) |
| 97 | \( 1 + (-2.93 - 1.96i)T + (37.1 + 89.6i)T^{2} \) |
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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
−11.22324334664292852777775110211, −10.59654907804780341776203784800, −9.930611992184315872899822682288, −8.502801898246531426849926055601, −7.29632546573905109189292773908, −6.50595544042296612891736021765, −5.43238541931759385284889283626, −4.51583627168172140029505090253, −3.64842404363081425605697113678, −2.77477626878152888001892637075,
1.01966101515442540884354689637, 2.14325055677961137670496414006, 3.84441114498515935010990693622, 5.03657675433086941299622707706, 5.72938824268671916373996656342, 6.85122407863860560896942102577, 7.55254127789030427424447653134, 8.479522460911814839138627762850, 9.718977355300893505762989227943, 11.20878291334037008210592182598
