L(s) = 1 | + (1.73 + i)2-s + (4.36 + 2.81i)3-s + (1.99 + 3.46i)4-s + (5.62 + 9.74i)5-s + (4.75 + 9.24i)6-s + (−17.3 + 6.56i)7-s + 7.99i·8-s + (11.1 + 24.5i)9-s + 22.5i·10-s + (−52.7 − 30.4i)11-s + (−1.00 + 20.7i)12-s + (73.7 − 42.5i)13-s + (−36.5 − 5.95i)14-s + (−2.83 + 58.4i)15-s + (−8 + 13.8i)16-s + 54.9·17-s + ⋯ |
L(s) = 1 | + (0.612 + 0.353i)2-s + (0.840 + 0.541i)3-s + (0.249 + 0.433i)4-s + (0.503 + 0.871i)5-s + (0.323 + 0.628i)6-s + (−0.935 + 0.354i)7-s + 0.353i·8-s + (0.413 + 0.910i)9-s + 0.711i·10-s + (−1.44 − 0.834i)11-s + (−0.0242 + 0.499i)12-s + (1.57 − 0.907i)13-s + (−0.697 − 0.113i)14-s + (−0.0488 + 1.00i)15-s + (−0.125 + 0.216i)16-s + 0.784·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0903 - 0.995i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.0903 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.94347 + 2.12770i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.94347 + 2.12770i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.73 - i)T \) |
| 3 | \( 1 + (-4.36 - 2.81i)T \) |
| 7 | \( 1 + (17.3 - 6.56i)T \) |
good | 5 | \( 1 + (-5.62 - 9.74i)T + (-62.5 + 108. i)T^{2} \) |
| 11 | \( 1 + (52.7 + 30.4i)T + (665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + (-73.7 + 42.5i)T + (1.09e3 - 1.90e3i)T^{2} \) |
| 17 | \( 1 - 54.9T + 4.91e3T^{2} \) |
| 19 | \( 1 - 9.90iT - 6.85e3T^{2} \) |
| 23 | \( 1 + (-6.77 + 3.91i)T + (6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-204. - 117. i)T + (1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 + (-53.1 + 30.6i)T + (1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + 218.T + 5.06e4T^{2} \) |
| 41 | \( 1 + (-36.9 - 63.9i)T + (-3.44e4 + 5.96e4i)T^{2} \) |
| 43 | \( 1 + (-116. + 201. i)T + (-3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 + (-43.8 + 76.0i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + 56.2iT - 1.48e5T^{2} \) |
| 59 | \( 1 + (-217. - 376. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (715. + 413. i)T + (1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (451. + 782. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + 894. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + 629. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + (469. - 812. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + (236. - 409. i)T + (-2.85e5 - 4.95e5i)T^{2} \) |
| 89 | \( 1 + 417.T + 7.04e5T^{2} \) |
| 97 | \( 1 + (485. + 280. i)T + (4.56e5 + 7.90e5i)T^{2} \) |
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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
−13.55605871185507243947930410303, −12.56102144930109685418103029703, −10.68177329432241461624590242522, −10.33421849341927278887979714412, −8.776272348302010430905947171554, −7.84415859831260357333881222880, −6.34364597297115242376631635487, −5.39290652286005596300672559477, −3.37808493636668900622230786269, −2.84623247353760332276223438820,
1.30631898179591294259270842614, 2.86311973287199477196284013608, 4.29091094161631363276908771444, 5.85340117601856134266401283908, 7.05694718664538135695524302789, 8.430290719899369384290634055777, 9.514811384314318750124075641360, 10.40426134932358077889771151916, 12.06480643694959596290350937920, 12.97227528672961237441688263584