L(s) = 1 | − 1.27i·2-s + (11.5 + 10.5i)3-s + 30.3·4-s + 80.0·5-s + (13.4 − 14.6i)6-s − 196. i·7-s − 79.5i·8-s + (21.7 + 242. i)9-s − 102. i·10-s − 20.8·11-s + (349. + 319. i)12-s − 791.·13-s − 251.·14-s + (921. + 842. i)15-s + 870.·16-s − 174.·17-s + ⋯ |
L(s) = 1 | − 0.225i·2-s + (0.738 + 0.674i)3-s + 0.949·4-s + 1.43·5-s + (0.152 − 0.166i)6-s − 1.51i·7-s − 0.439i·8-s + (0.0895 + 0.995i)9-s − 0.323i·10-s − 0.0520·11-s + (0.700 + 0.640i)12-s − 1.29·13-s − 0.342·14-s + (1.05 + 0.966i)15-s + 0.849·16-s − 0.146·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.103i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.994 + 0.103i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(3.18736 - 0.165164i\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.18736 - 0.165164i\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-11.5 - 10.5i)T \) |
| 23 | \( 1 + (2.03e3 - 1.50e3i)T \) |
good | 2 | \( 1 + 1.27iT - 32T^{2} \) |
| 5 | \( 1 - 80.0T + 3.12e3T^{2} \) |
| 7 | \( 1 + 196. iT - 1.68e4T^{2} \) |
| 11 | \( 1 + 20.8T + 1.61e5T^{2} \) |
| 13 | \( 1 + 791.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 174.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 892. iT - 2.47e6T^{2} \) |
| 29 | \( 1 - 5.42e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 - 3.74e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.07e4iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 5.10e3iT - 1.15e8T^{2} \) |
| 43 | \( 1 - 1.01e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 2.20e3iT - 2.29e8T^{2} \) |
| 53 | \( 1 + 3.84e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 4.96e4iT - 7.14e8T^{2} \) |
| 61 | \( 1 + 2.88e4iT - 8.44e8T^{2} \) |
| 67 | \( 1 - 6.11e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 1.17e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 - 4.87e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 6.58e4iT - 3.07e9T^{2} \) |
| 83 | \( 1 + 5.35e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.06e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 5.96e4iT - 8.58e9T^{2} \) |
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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
−13.88278277957163200612701489644, −12.89577447172178805084496798248, −11.12349050673604539192161595015, −10.09660603643535186029136362239, −9.695764072024392698303380256659, −7.76493678551749398876658795277, −6.61677796312516842602693616930, −4.83787532711420772432468012751, −3.06146780199036450282808854219, −1.71397590057805465368091996023,
2.06608418948565208080380634311, 2.50730459215817265272718088891, 5.63772955594262582035195240164, 6.42122590168278128291038898778, 7.84027909545017324998373062363, 9.116932379047208731097729163472, 10.10540556923653460811062364693, 11.96134183834443403124228929072, 12.55761923273117671410011721560, 13.95890464186692065724898606351