| L(s) = 1 | + (0.635 + 1.89i)2-s + (−1.35 − 1.07i)3-s + (−3.19 + 2.40i)4-s + (0.639 + 0.308i)5-s + (1.18 − 3.25i)6-s + (−5.65 − 4.50i)7-s + (−6.59 − 4.52i)8-s + (0.667 + 2.92i)9-s + (−0.177 + 1.40i)10-s + (5.38 + 1.22i)11-s + (6.92 + 0.185i)12-s + (0.952 − 4.17i)13-s + (4.95 − 13.5i)14-s + (−0.533 − 1.10i)15-s + (4.39 − 15.3i)16-s + 29.9·17-s + ⋯ |
| L(s) = 1 | + (0.317 + 0.948i)2-s + (−0.451 − 0.359i)3-s + (−0.798 + 0.602i)4-s + (0.127 + 0.0616i)5-s + (0.197 − 0.542i)6-s + (−0.807 − 0.644i)7-s + (−0.824 − 0.565i)8-s + (0.0741 + 0.324i)9-s + (−0.0177 + 0.140i)10-s + (0.489 + 0.111i)11-s + (0.577 + 0.0154i)12-s + (0.0732 − 0.321i)13-s + (0.354 − 0.970i)14-s + (−0.0355 − 0.0738i)15-s + (0.274 − 0.961i)16-s + 1.75·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 348 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.975 + 0.218i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 348 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.975 + 0.218i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.26497 - 0.139581i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.26497 - 0.139581i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.635 - 1.89i)T \) |
| 3 | \( 1 + (1.35 + 1.07i)T \) |
| 29 | \( 1 + (21.4 + 19.4i)T \) |
| good | 5 | \( 1 + (-0.639 - 0.308i)T + (15.5 + 19.5i)T^{2} \) |
| 7 | \( 1 + (5.65 + 4.50i)T + (10.9 + 47.7i)T^{2} \) |
| 11 | \( 1 + (-5.38 - 1.22i)T + (109. + 52.4i)T^{2} \) |
| 13 | \( 1 + (-0.952 + 4.17i)T + (-152. - 73.3i)T^{2} \) |
| 17 | \( 1 - 29.9T + 289T^{2} \) |
| 19 | \( 1 + (-18.0 + 14.4i)T + (80.3 - 351. i)T^{2} \) |
| 23 | \( 1 + (7.70 + 16.0i)T + (-329. + 413. i)T^{2} \) |
| 31 | \( 1 + (-3.20 + 6.66i)T + (-599. - 751. i)T^{2} \) |
| 37 | \( 1 + (-3.34 - 14.6i)T + (-1.23e3 + 593. i)T^{2} \) |
| 41 | \( 1 - 56.4T + 1.68e3T^{2} \) |
| 43 | \( 1 + (-18.2 - 37.9i)T + (-1.15e3 + 1.44e3i)T^{2} \) |
| 47 | \( 1 + (-39.3 - 8.98i)T + (1.99e3 + 958. i)T^{2} \) |
| 53 | \( 1 + (71.5 + 34.4i)T + (1.75e3 + 2.19e3i)T^{2} \) |
| 59 | \( 1 + 98.4iT - 3.48e3T^{2} \) |
| 61 | \( 1 + (13.9 - 17.4i)T + (-828. - 3.62e3i)T^{2} \) |
| 67 | \( 1 + (-70.6 + 16.1i)T + (4.04e3 - 1.94e3i)T^{2} \) |
| 71 | \( 1 + (88.1 + 20.1i)T + (4.54e3 + 2.18e3i)T^{2} \) |
| 73 | \( 1 + (70.1 - 33.7i)T + (3.32e3 - 4.16e3i)T^{2} \) |
| 79 | \( 1 + (21.9 - 5.00i)T + (5.62e3 - 2.70e3i)T^{2} \) |
| 83 | \( 1 + (-18.0 + 14.4i)T + (1.53e3 - 6.71e3i)T^{2} \) |
| 89 | \( 1 + (142. + 68.5i)T + (4.93e3 + 6.19e3i)T^{2} \) |
| 97 | \( 1 + (3.41 + 4.28i)T + (-2.09e3 + 9.17e3i)T^{2} \) |
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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
−11.46401773298954701580783860672, −10.03850442252291433592328456547, −9.486299143117888451585581597287, −8.015131551128886946174493685336, −7.34541782326372310336507071476, −6.35107111530175933646530110179, −5.65244354115415932884810613667, −4.34161405107263687911497892925, −3.16178377433693013300343271598, −0.63952899537111621702760963093,
1.34931861102946636391889260492, 3.11790649138901709071013264373, 3.95265740202853138282750929225, 5.58561139470317621632164867225, 5.82122551881111769088408015839, 7.53754269069470688556795349152, 9.119461577446530789462038609124, 9.556711125505932789018634811286, 10.39016363461848941794023610788, 11.48612137516132754793167554641
