L(s) = 1 | + (−0.987 + 2.65i)2-s + (2.12 + 2.12i)3-s + (−6.05 − 5.23i)4-s + (−11.8 + 11.8i)5-s + (−7.71 + 3.52i)6-s + 0.485i·7-s + (19.8 − 10.8i)8-s + 8.99i·9-s + (−19.7 − 43.2i)10-s + (−30.9 + 30.9i)11-s + (−1.73 − 23.9i)12-s + (−18.4 − 18.4i)13-s + (−1.28 − 0.478i)14-s − 50.4·15-s + (9.24 + 63.3i)16-s + 135.·17-s + ⋯ |
L(s) = 1 | + (−0.348 + 0.937i)2-s + (0.408 + 0.408i)3-s + (−0.756 − 0.654i)4-s + (−1.06 + 1.06i)5-s + (−0.525 + 0.240i)6-s + 0.0261i·7-s + (0.876 − 0.480i)8-s + 0.333i·9-s + (−0.625 − 1.36i)10-s + (−0.848 + 0.848i)11-s + (−0.0418 − 0.575i)12-s + (−0.393 − 0.393i)13-s + (−0.0245 − 0.00913i)14-s − 0.868·15-s + (0.144 + 0.989i)16-s + 1.93·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.969 - 0.245i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.969 - 0.245i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.105598 + 0.848036i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.105598 + 0.848036i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.987 - 2.65i)T \) |
| 3 | \( 1 + (-2.12 - 2.12i)T \) |
good | 5 | \( 1 + (11.8 - 11.8i)T - 125iT^{2} \) |
| 7 | \( 1 - 0.485iT - 343T^{2} \) |
| 11 | \( 1 + (30.9 - 30.9i)T - 1.33e3iT^{2} \) |
| 13 | \( 1 + (18.4 + 18.4i)T + 2.19e3iT^{2} \) |
| 17 | \( 1 - 135.T + 4.91e3T^{2} \) |
| 19 | \( 1 + (-65.8 - 65.8i)T + 6.85e3iT^{2} \) |
| 23 | \( 1 - 128. iT - 1.21e4T^{2} \) |
| 29 | \( 1 + (-6.64 - 6.64i)T + 2.43e4iT^{2} \) |
| 31 | \( 1 + 15.1T + 2.97e4T^{2} \) |
| 37 | \( 1 + (-51.4 + 51.4i)T - 5.06e4iT^{2} \) |
| 41 | \( 1 + 410. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + (-69.9 + 69.9i)T - 7.95e4iT^{2} \) |
| 47 | \( 1 + 487.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (217. - 217. i)T - 1.48e5iT^{2} \) |
| 59 | \( 1 + (293. - 293. i)T - 2.05e5iT^{2} \) |
| 61 | \( 1 + (-207. - 207. i)T + 2.26e5iT^{2} \) |
| 67 | \( 1 + (-284. - 284. i)T + 3.00e5iT^{2} \) |
| 71 | \( 1 + 614. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + 486. iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 960.T + 4.93e5T^{2} \) |
| 83 | \( 1 + (-463. - 463. i)T + 5.71e5iT^{2} \) |
| 89 | \( 1 - 1.27e3iT - 7.04e5T^{2} \) |
| 97 | \( 1 + 994.T + 9.12e5T^{2} \) |
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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
−15.49167361899917733425893422717, −14.90196656215938214689017137763, −13.97546634127273360763490435963, −12.23313171087944658663803112070, −10.55314701983827039762430300204, −9.717402828204906082732004496706, −7.79538820190796973048656061741, −7.46431480071737310064691004228, −5.38501224850758983232648874951, −3.53449507208014420266043670009,
0.74975134496162749860328636908, 3.20412086605477390798507592002, 4.89477892226352064745948975228, 7.74453018355791605387131426259, 8.408529399511264928667591913206, 9.729963745312137024038126635148, 11.34893853850718707037356484901, 12.27663273529408127941419664379, 13.09745019207039843711895061428, 14.35775294754145409454105382426