L(s) = 1 | + 10.3i·2-s + (−12.5 + 9.28i)3-s − 74.8·4-s − 31.7·5-s + (−95.9 − 129. i)6-s − 442. i·8-s + (70.5 − 232. i)9-s − 328. i·10-s + 453. i·11-s + (936. − 694. i)12-s − 551. i·13-s + (397. − 294. i)15-s + 2.17e3·16-s − 538.·17-s + (2.40e3 + 729. i)18-s − 1.36e3i·19-s + ⋯ |
L(s) = 1 | + 1.82i·2-s + (−0.803 + 0.595i)3-s − 2.33·4-s − 0.567·5-s + (−1.08 − 1.46i)6-s − 2.44i·8-s + (0.290 − 0.956i)9-s − 1.03i·10-s + 1.13i·11-s + (1.87 − 1.39i)12-s − 0.905i·13-s + (0.456 − 0.338i)15-s + 2.12·16-s − 0.451·17-s + (1.74 + 0.530i)18-s − 0.868i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.217 - 0.976i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.217 - 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.5313411305\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5313411305\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (12.5 - 9.28i)T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 - 10.3iT - 32T^{2} \) |
| 5 | \( 1 + 31.7T + 3.12e3T^{2} \) |
| 11 | \( 1 - 453. iT - 1.61e5T^{2} \) |
| 13 | \( 1 + 551. iT - 3.71e5T^{2} \) |
| 17 | \( 1 + 538.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.36e3iT - 2.47e6T^{2} \) |
| 23 | \( 1 - 3.23e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 - 1.60e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 + 7.06e3iT - 2.86e7T^{2} \) |
| 37 | \( 1 + 1.95e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.80e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 7.88e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 6.09e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.41e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 1.69e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.97e4iT - 8.44e8T^{2} \) |
| 67 | \( 1 - 1.36e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 3.13e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 - 9.82e3iT - 2.07e9T^{2} \) |
| 79 | \( 1 - 9.97e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 7.95e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 953.T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.15e5iT - 8.58e9T^{2} \) |
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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
−12.57477341197986608164959010126, −11.34751872872604463567076970878, −9.961680068819355881677297118574, −9.120961892407753278405048952901, −7.77253042544667725547984759973, −7.01361546466568425506263423041, −5.83392198483638772414053487539, −4.91004240339845193917205874033, −3.94404910078529808534404903652, −0.29930013478146209963840675260,
0.832674219071741408468897922899, 2.14406737068596101996830743370, 3.68332608446943822496509106372, 4.82183394220627693554926840866, 6.31962082367769562425772161654, 8.014005277357732871759961625307, 9.049596648911055703771752507571, 10.46807015679668845033739692442, 11.04436627261037569694901174507, 11.93575120257120289583408339532