| L(s) = 1 | + (−2.22 + 1.03i)2-s + (1.24 + 1.20i)3-s + (2.58 − 3.08i)4-s + (2.78 − 1.95i)5-s + (−4.01 − 1.38i)6-s + (0.634 − 3.60i)7-s + (−1.28 + 4.79i)8-s + (0.0984 + 2.99i)9-s + (−4.17 + 7.22i)10-s + (0.183 + 0.317i)11-s + (6.93 − 0.721i)12-s + (−1.83 + 0.160i)13-s + (2.32 + 8.66i)14-s + (5.81 + 0.927i)15-s + (−0.719 − 4.07i)16-s + (−2.38 − 0.208i)17-s + ⋯ |
| L(s) = 1 | + (−1.57 + 0.733i)2-s + (0.718 + 0.695i)3-s + (1.29 − 1.54i)4-s + (1.24 − 0.872i)5-s + (−1.64 − 0.566i)6-s + (0.239 − 1.36i)7-s + (−0.454 + 1.69i)8-s + (0.0328 + 0.999i)9-s + (−1.31 + 2.28i)10-s + (0.0553 + 0.0958i)11-s + (2.00 − 0.208i)12-s + (−0.509 + 0.0445i)13-s + (0.620 + 2.31i)14-s + (1.50 + 0.239i)15-s + (−0.179 − 1.01i)16-s + (−0.578 − 0.0505i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 111 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.803 - 0.595i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 111 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.803 - 0.595i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.709704 + 0.234283i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.709704 + 0.234283i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.24 - 1.20i)T \) |
| 37 | \( 1 + (-5.77 + 1.90i)T \) |
| good | 2 | \( 1 + (2.22 - 1.03i)T + (1.28 - 1.53i)T^{2} \) |
| 5 | \( 1 + (-2.78 + 1.95i)T + (1.71 - 4.69i)T^{2} \) |
| 7 | \( 1 + (-0.634 + 3.60i)T + (-6.57 - 2.39i)T^{2} \) |
| 11 | \( 1 + (-0.183 - 0.317i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (1.83 - 0.160i)T + (12.8 - 2.25i)T^{2} \) |
| 17 | \( 1 + (2.38 + 0.208i)T + (16.7 + 2.95i)T^{2} \) |
| 19 | \( 1 + (2.29 - 4.92i)T + (-12.2 - 14.5i)T^{2} \) |
| 23 | \( 1 + (3.68 - 0.986i)T + (19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 + (1.48 + 0.398i)T + (25.1 + 14.5i)T^{2} \) |
| 31 | \( 1 + (-2.30 - 2.30i)T + 31iT^{2} \) |
| 41 | \( 1 + (-8.83 - 7.41i)T + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (7.26 - 7.26i)T - 43iT^{2} \) |
| 47 | \( 1 + (2.81 + 1.62i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-2.57 + 0.453i)T + (49.8 - 18.1i)T^{2} \) |
| 59 | \( 1 + (2.38 - 3.40i)T + (-20.1 - 55.4i)T^{2} \) |
| 61 | \( 1 + (1.18 + 13.5i)T + (-60.0 + 10.5i)T^{2} \) |
| 67 | \( 1 + (6.97 + 1.22i)T + (62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (0.0966 + 0.265i)T + (-54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + 4.80iT - 73T^{2} \) |
| 79 | \( 1 + (0.0530 + 0.0757i)T + (-27.0 + 74.2i)T^{2} \) |
| 83 | \( 1 + (2.90 + 3.46i)T + (-14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (2.77 + 1.94i)T + (30.4 + 83.6i)T^{2} \) |
| 97 | \( 1 + (-0.864 - 3.22i)T + (-84.0 + 48.5i)T^{2} \) |
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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
−14.05626337759612640729726673410, −13.06613446324030424292246293938, −10.91111180227337242457782583094, −9.936608904619229762700692389442, −9.604652959455128264833196446658, −8.436015167292884824379771568305, −7.60501652801684614461002351298, −6.13227319028976817426047210520, −4.53637045843372381451800266266, −1.71609162690878312444799828846,
2.15543329023922101474615052987, 2.61682930439847241152505364361, 6.09183752820405027618167947869, 7.22235946364293597378880200193, 8.551623614656902183802714735229, 9.207668896181967221198157009562, 10.07896153345638732983186835824, 11.27579127599730029538273403334, 12.25150809733206197930148591971, 13.37975359881385289662449860182
