| L(s) = 1 | + (−1.77 − 1.53i)3-s + (−3.72 − 0.536i)5-s + (−1.80 − 3.95i)7-s + (0.356 + 2.48i)9-s + (−0.791 − 2.69i)11-s + (−1.31 − 0.602i)13-s + (5.79 + 6.68i)15-s + (0.426 − 0.274i)17-s + (1.09 − 1.69i)19-s + (−2.87 + 9.79i)21-s + (−4.04 − 2.57i)23-s + (8.82 + 2.59i)25-s + (−0.626 + 0.974i)27-s + (0.637 + 0.991i)29-s + (5.28 + 6.09i)31-s + ⋯ |
| L(s) = 1 | + (−1.02 − 0.887i)3-s + (−1.66 − 0.239i)5-s + (−0.682 − 1.49i)7-s + (0.118 + 0.826i)9-s + (−0.238 − 0.812i)11-s + (−0.365 − 0.167i)13-s + (1.49 + 1.72i)15-s + (0.103 − 0.0664i)17-s + (0.250 − 0.389i)19-s + (−0.627 + 2.13i)21-s + (−0.843 − 0.537i)23-s + (1.76 + 0.518i)25-s + (−0.120 + 0.187i)27-s + (0.118 + 0.184i)29-s + (0.948 + 1.09i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0898 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0898 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.131180 + 0.119885i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.131180 + 0.119885i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 23 | \( 1 + (4.04 + 2.57i)T \) |
| good | 3 | \( 1 + (1.77 + 1.53i)T + (0.426 + 2.96i)T^{2} \) |
| 5 | \( 1 + (3.72 + 0.536i)T + (4.79 + 1.40i)T^{2} \) |
| 7 | \( 1 + (1.80 + 3.95i)T + (-4.58 + 5.29i)T^{2} \) |
| 11 | \( 1 + (0.791 + 2.69i)T + (-9.25 + 5.94i)T^{2} \) |
| 13 | \( 1 + (1.31 + 0.602i)T + (8.51 + 9.82i)T^{2} \) |
| 17 | \( 1 + (-0.426 + 0.274i)T + (7.06 - 15.4i)T^{2} \) |
| 19 | \( 1 + (-1.09 + 1.69i)T + (-7.89 - 17.2i)T^{2} \) |
| 29 | \( 1 + (-0.637 - 0.991i)T + (-12.0 + 26.3i)T^{2} \) |
| 31 | \( 1 + (-5.28 - 6.09i)T + (-4.41 + 30.6i)T^{2} \) |
| 37 | \( 1 + (-1.78 + 0.256i)T + (35.5 - 10.4i)T^{2} \) |
| 41 | \( 1 + (-1.06 + 7.38i)T + (-39.3 - 11.5i)T^{2} \) |
| 43 | \( 1 + (0.318 + 0.275i)T + (6.11 + 42.5i)T^{2} \) |
| 47 | \( 1 + 11.5T + 47T^{2} \) |
| 53 | \( 1 + (-8.99 + 4.10i)T + (34.7 - 40.0i)T^{2} \) |
| 59 | \( 1 + (-1.14 - 0.524i)T + (38.6 + 44.5i)T^{2} \) |
| 61 | \( 1 + (-6.78 + 5.87i)T + (8.68 - 60.3i)T^{2} \) |
| 67 | \( 1 + (0.439 - 1.49i)T + (-56.3 - 36.2i)T^{2} \) |
| 71 | \( 1 + (-3.81 - 1.11i)T + (59.7 + 38.3i)T^{2} \) |
| 73 | \( 1 + (3.83 + 2.46i)T + (30.3 + 66.4i)T^{2} \) |
| 79 | \( 1 + (-4.36 + 9.55i)T + (-51.7 - 59.7i)T^{2} \) |
| 83 | \( 1 + (9.43 - 1.35i)T + (79.6 - 23.3i)T^{2} \) |
| 89 | \( 1 + (7.96 - 9.18i)T + (-12.6 - 88.0i)T^{2} \) |
| 97 | \( 1 + (-0.199 + 1.38i)T + (-93.0 - 27.3i)T^{2} \) |
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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
−10.03638317015041755657704712866, −8.530393130881586752296716619791, −7.76082988719328494703832179962, −7.05707386135175553356778262800, −6.49869172275637334905457188856, −5.18300158363997688265868677154, −4.12370786266505322403520344016, −3.25331792493018868896030840503, −0.858027694510186982901172801478, −0.14951409521198220197240210882,
2.65264150373558351490715293052, 3.87094013164283471183454689005, 4.66394080900347149034541251878, 5.61117498163913192512197043569, 6.48781740729318107341647631692, 7.64540405537111862026045627857, 8.406337713806231972298096438936, 9.664700208794213624021198352060, 10.05441427741680290599608144722, 11.31536021875867677122755658168
