| L(s) = 1 | − 1.15i·3-s + (−1.04 − 3.22i)5-s + (−0.587 − 0.809i)7-s + 1.66·9-s + (−4.79 − 1.55i)11-s + (−0.242 + 0.334i)13-s + (−3.72 + 1.21i)15-s + (4.09 + 1.33i)17-s + (−4.03 − 5.55i)19-s + (−0.934 + 0.679i)21-s + (2.71 + 1.97i)23-s + (−5.27 + 3.83i)25-s − 5.38i·27-s + (−5.54 + 1.80i)29-s + (1.17 − 3.62i)31-s + ⋯ |
| L(s) = 1 | − 0.667i·3-s + (−0.469 − 1.44i)5-s + (−0.222 − 0.305i)7-s + 0.555·9-s + (−1.44 − 0.469i)11-s + (−0.0673 + 0.0926i)13-s + (−0.962 + 0.312i)15-s + (0.994 + 0.323i)17-s + (−0.925 − 1.27i)19-s + (−0.203 + 0.148i)21-s + (0.565 + 0.411i)23-s + (−1.05 + 0.766i)25-s − 1.03i·27-s + (−1.02 + 0.334i)29-s + (0.211 − 0.651i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.964 - 0.265i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.964 - 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8492487005\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8492487005\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (0.587 + 0.809i)T \) |
| 41 | \( 1 + (2.10 - 6.04i)T \) |
| good | 3 | \( 1 + 1.15iT - 3T^{2} \) |
| 5 | \( 1 + (1.04 + 3.22i)T + (-4.04 + 2.93i)T^{2} \) |
| 11 | \( 1 + (4.79 + 1.55i)T + (8.89 + 6.46i)T^{2} \) |
| 13 | \( 1 + (0.242 - 0.334i)T + (-4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (-4.09 - 1.33i)T + (13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (4.03 + 5.55i)T + (-5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (-2.71 - 1.97i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (5.54 - 1.80i)T + (23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (-1.17 + 3.62i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (-1.51 - 4.66i)T + (-29.9 + 21.7i)T^{2} \) |
| 43 | \( 1 + (-2.08 - 1.51i)T + (13.2 + 40.8i)T^{2} \) |
| 47 | \( 1 + (2.14 - 2.95i)T + (-14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (4.59 - 1.49i)T + (42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (-2.76 - 2.01i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-9.57 + 6.95i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (4.00 - 1.29i)T + (54.2 - 39.3i)T^{2} \) |
| 71 | \( 1 + (3.48 + 1.13i)T + (57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + 16.2T + 73T^{2} \) |
| 79 | \( 1 + 5.78iT - 79T^{2} \) |
| 83 | \( 1 + 8.92T + 83T^{2} \) |
| 89 | \( 1 + (-4.55 - 6.27i)T + (-27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (-0.934 + 0.303i)T + (78.4 - 57.0i)T^{2} \) |
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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
−9.271909931470701762712634129351, −8.321081285766953517945855951232, −7.83352138402049233501585465312, −7.05863138047132206546233108721, −5.90448140135982502180260390585, −4.98102255637828273689340509069, −4.27610255267367529961714923234, −2.94410669678658106201467467412, −1.48730829381584848976159089642, −0.37113180111642116465216746096,
2.20979570336403735782583816874, 3.20672860720128179308816669343, 3.97475535072281779298027367588, 5.12315002740952439594703437468, 6.00346427531541071116108220344, 7.18496940133013672483343648116, 7.53720537856903191443699179869, 8.579280770235756147118178898660, 9.801430977547135119541431216998, 10.34119072612543180454975866843
