| L(s) = 1 | + (−0.142 + 0.989i)2-s + (1.27 + 2.79i)3-s + (−0.959 − 0.281i)4-s + (0.685 + 0.791i)5-s + (−2.94 + 0.865i)6-s + (−0.841 − 0.540i)7-s + (0.415 − 0.909i)8-s + (−4.21 + 4.85i)9-s + (−0.880 + 0.566i)10-s + (0.0338 + 0.235i)11-s + (−0.437 − 3.03i)12-s + (0.199 − 0.128i)13-s + (0.654 − 0.755i)14-s + (−1.33 + 2.92i)15-s + (0.841 + 0.540i)16-s + (0.132 − 0.0389i)17-s + ⋯ |
| L(s) = 1 | + (−0.100 + 0.699i)2-s + (0.736 + 1.61i)3-s + (−0.479 − 0.140i)4-s + (0.306 + 0.353i)5-s + (−1.20 + 0.353i)6-s + (−0.317 − 0.204i)7-s + (0.146 − 0.321i)8-s + (−1.40 + 1.61i)9-s + (−0.278 + 0.179i)10-s + (0.0102 + 0.0709i)11-s + (−0.126 − 0.877i)12-s + (0.0554 − 0.0356i)13-s + (0.175 − 0.201i)14-s + (−0.344 + 0.755i)15-s + (0.210 + 0.135i)16-s + (0.0321 − 0.00944i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.924 - 0.380i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.924 - 0.380i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.282864 + 1.43180i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.282864 + 1.43180i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.142 - 0.989i)T \) |
| 7 | \( 1 + (0.841 + 0.540i)T \) |
| 23 | \( 1 + (-3.62 - 3.13i)T \) |
| good | 3 | \( 1 + (-1.27 - 2.79i)T + (-1.96 + 2.26i)T^{2} \) |
| 5 | \( 1 + (-0.685 - 0.791i)T + (-0.711 + 4.94i)T^{2} \) |
| 11 | \( 1 + (-0.0338 - 0.235i)T + (-10.5 + 3.09i)T^{2} \) |
| 13 | \( 1 + (-0.199 + 0.128i)T + (5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (-0.132 + 0.0389i)T + (14.3 - 9.19i)T^{2} \) |
| 19 | \( 1 + (-1.12 - 0.330i)T + (15.9 + 10.2i)T^{2} \) |
| 29 | \( 1 + (-2.08 + 0.611i)T + (24.3 - 15.6i)T^{2} \) |
| 31 | \( 1 + (-3.55 + 7.77i)T + (-20.3 - 23.4i)T^{2} \) |
| 37 | \( 1 + (7.11 - 8.20i)T + (-5.26 - 36.6i)T^{2} \) |
| 41 | \( 1 + (3.94 + 4.55i)T + (-5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (-1.06 - 2.34i)T + (-28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 - 8.37T + 47T^{2} \) |
| 53 | \( 1 + (-8.46 - 5.44i)T + (22.0 + 48.2i)T^{2} \) |
| 59 | \( 1 + (-2.31 + 1.48i)T + (24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (3.25 - 7.12i)T + (-39.9 - 46.1i)T^{2} \) |
| 67 | \( 1 + (0.867 - 6.03i)T + (-64.2 - 18.8i)T^{2} \) |
| 71 | \( 1 + (-1.19 + 8.33i)T + (-68.1 - 20.0i)T^{2} \) |
| 73 | \( 1 + (13.9 + 4.10i)T + (61.4 + 39.4i)T^{2} \) |
| 79 | \( 1 + (-11.0 + 7.08i)T + (32.8 - 71.8i)T^{2} \) |
| 83 | \( 1 + (-2.61 + 3.01i)T + (-11.8 - 82.1i)T^{2} \) |
| 89 | \( 1 + (-0.706 - 1.54i)T + (-58.2 + 67.2i)T^{2} \) |
| 97 | \( 1 + (3.29 + 3.80i)T + (-13.8 + 96.0i)T^{2} \) |
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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
−11.86012116995140588376326943535, −10.54647014546673828655811735366, −10.09503013104676245786688491515, −9.232203096196939695352444395977, −8.491320600520295703663537303082, −7.36738096875057528730707279606, −6.05336459830819692081428425557, −4.94634125072127049503701239262, −3.95602204221433045252205723118, −2.83562247682245101302351566709,
1.10055258060545571121997253720, 2.34112495576779253356256254659, 3.40171745857496699968451153376, 5.27129384865596309941259321572, 6.58120583438380604753792514818, 7.42528621591338188224200496662, 8.655759279700778162258959650799, 8.971399179246837814817176384637, 10.28424805595667140467125505821, 11.50961690929097712697419280302
