| L(s) = 1 | + (1.74 − 0.974i)2-s + (2.87 − 0.862i)3-s + (2.10 − 3.40i)4-s + (4.31 + 2.52i)5-s + (4.17 − 4.30i)6-s + (−3.94 + 3.94i)7-s + (0.355 − 7.99i)8-s + (7.51 − 4.95i)9-s + (9.99 + 0.212i)10-s + (3.09 + 2.24i)11-s + (3.10 − 11.5i)12-s + (−1.77 + 11.2i)13-s + (−3.04 + 10.7i)14-s + (14.5 + 3.54i)15-s + (−7.16 − 14.3i)16-s + (−5.54 − 10.8i)17-s + ⋯ |
| L(s) = 1 | + (0.873 − 0.487i)2-s + (0.957 − 0.287i)3-s + (0.525 − 0.850i)4-s + (0.862 + 0.505i)5-s + (0.696 − 0.717i)6-s + (−0.563 + 0.563i)7-s + (0.0443 − 0.999i)8-s + (0.834 − 0.550i)9-s + (0.999 + 0.0212i)10-s + (0.281 + 0.204i)11-s + (0.258 − 0.965i)12-s + (−0.136 + 0.863i)13-s + (−0.217 + 0.766i)14-s + (0.971 + 0.236i)15-s + (−0.447 − 0.894i)16-s + (−0.326 − 0.639i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.682 + 0.730i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.682 + 0.730i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.57819 - 1.55326i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.57819 - 1.55326i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.74 + 0.974i)T \) |
| 3 | \( 1 + (-2.87 + 0.862i)T \) |
| 5 | \( 1 + (-4.31 - 2.52i)T \) |
| good | 7 | \( 1 + (3.94 - 3.94i)T - 49iT^{2} \) |
| 11 | \( 1 + (-3.09 - 2.24i)T + (37.3 + 115. i)T^{2} \) |
| 13 | \( 1 + (1.77 - 11.2i)T + (-160. - 52.2i)T^{2} \) |
| 17 | \( 1 + (5.54 + 10.8i)T + (-169. + 233. i)T^{2} \) |
| 19 | \( 1 + (-5.87 + 18.0i)T + (-292. - 212. i)T^{2} \) |
| 23 | \( 1 + (44.6 - 7.07i)T + (503. - 163. i)T^{2} \) |
| 29 | \( 1 + (-11.5 - 35.6i)T + (-680. + 494. i)T^{2} \) |
| 31 | \( 1 + (-1.42 - 0.463i)T + (777. + 564. i)T^{2} \) |
| 37 | \( 1 + (50.6 + 8.02i)T + (1.30e3 + 423. i)T^{2} \) |
| 41 | \( 1 + (44.7 + 61.5i)T + (-519. + 1.59e3i)T^{2} \) |
| 43 | \( 1 + (-28.5 - 28.5i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (7.19 - 14.1i)T + (-1.29e3 - 1.78e3i)T^{2} \) |
| 53 | \( 1 + (-39.6 + 77.8i)T + (-1.65e3 - 2.27e3i)T^{2} \) |
| 59 | \( 1 + (-19.0 - 26.2i)T + (-1.07e3 + 3.31e3i)T^{2} \) |
| 61 | \( 1 + (-17.5 - 12.7i)T + (1.14e3 + 3.53e3i)T^{2} \) |
| 67 | \( 1 + (-42.9 - 84.3i)T + (-2.63e3 + 3.63e3i)T^{2} \) |
| 71 | \( 1 + (12.0 + 37.0i)T + (-4.07e3 + 2.96e3i)T^{2} \) |
| 73 | \( 1 + (-21.6 + 3.43i)T + (5.06e3 - 1.64e3i)T^{2} \) |
| 79 | \( 1 + (-7.72 - 23.7i)T + (-5.04e3 + 3.66e3i)T^{2} \) |
| 83 | \( 1 + (-35.9 - 70.5i)T + (-4.04e3 + 5.57e3i)T^{2} \) |
| 89 | \( 1 + (62.9 + 45.7i)T + (2.44e3 + 7.53e3i)T^{2} \) |
| 97 | \( 1 + (21.0 - 41.2i)T + (-5.53e3 - 7.61e3i)T^{2} \) |
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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
−11.68474082750215979672793234330, −10.36204150984701001606300168077, −9.574234044294791294477787680883, −8.915991817650253317210523651458, −7.07516932166516033519753962117, −6.56670601118786524104264932406, −5.29405022785504908812019481347, −3.84197838924679759170927920048, −2.69641482311009404763872641669, −1.84817737050573569869835214999,
2.03301743573609118591977193097, 3.42305097148881780830262918469, 4.34460609957725051447971530791, 5.65897734808545688449580263577, 6.57491969631641291383711368510, 7.907541458325007615169709391255, 8.520046377459660607817764729892, 9.879491001831362788874551714643, 10.39067407403528606681279471227, 12.08816959667301698915162741428
