| L(s) = 1 | + (−0.691 + 1.90i)2-s + (1.70 − 0.300i)3-s + (−0.0681 − 0.0571i)4-s + (6.34 − 5.32i)5-s + (−0.608 + 3.44i)6-s + (−5.23 + 9.07i)7-s + (−6.84 + 3.95i)8-s + (2.81 − 1.02i)9-s + (5.72 + 15.7i)10-s + (−6.84 − 11.8i)11-s + (−0.133 − 0.0770i)12-s + (4.21 + 0.743i)13-s + (−13.6 − 16.2i)14-s + (9.21 − 10.9i)15-s + (−2.83 − 16.0i)16-s + (16.6 + 6.07i)17-s + ⋯ |
| L(s) = 1 | + (−0.345 + 0.950i)2-s + (0.568 − 0.100i)3-s + (−0.0170 − 0.0142i)4-s + (1.26 − 1.06i)5-s + (−0.101 + 0.574i)6-s + (−0.748 + 1.29i)7-s + (−0.856 + 0.494i)8-s + (0.313 − 0.114i)9-s + (0.572 + 1.57i)10-s + (−0.622 − 1.07i)11-s + (−0.0111 − 0.00641i)12-s + (0.324 + 0.0571i)13-s + (−0.972 − 1.15i)14-s + (0.614 − 0.732i)15-s + (−0.177 − 1.00i)16-s + (0.981 + 0.357i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.475 - 0.879i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.475 - 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.12272 + 0.669187i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.12272 + 0.669187i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.70 + 0.300i)T \) |
| 19 | \( 1 + (17.6 + 6.96i)T \) |
| good | 2 | \( 1 + (0.691 - 1.90i)T + (-3.06 - 2.57i)T^{2} \) |
| 5 | \( 1 + (-6.34 + 5.32i)T + (4.34 - 24.6i)T^{2} \) |
| 7 | \( 1 + (5.23 - 9.07i)T + (-24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (6.84 + 11.8i)T + (-60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (-4.21 - 0.743i)T + (158. + 57.8i)T^{2} \) |
| 17 | \( 1 + (-16.6 - 6.07i)T + (221. + 185. i)T^{2} \) |
| 23 | \( 1 + (25.9 + 21.7i)T + (91.8 + 520. i)T^{2} \) |
| 29 | \( 1 + (-0.352 - 0.968i)T + (-644. + 540. i)T^{2} \) |
| 31 | \( 1 + (-13.2 - 7.64i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 - 37.7iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (24.9 - 4.40i)T + (1.57e3 - 574. i)T^{2} \) |
| 43 | \( 1 + (-18.9 + 15.9i)T + (321. - 1.82e3i)T^{2} \) |
| 47 | \( 1 + (-30.1 + 10.9i)T + (1.69e3 - 1.41e3i)T^{2} \) |
| 53 | \( 1 + (13.9 - 16.5i)T + (-487. - 2.76e3i)T^{2} \) |
| 59 | \( 1 + (17.0 - 46.7i)T + (-2.66e3 - 2.23e3i)T^{2} \) |
| 61 | \( 1 + (-33.6 - 28.2i)T + (646. + 3.66e3i)T^{2} \) |
| 67 | \( 1 + (-30.4 - 83.5i)T + (-3.43e3 + 2.88e3i)T^{2} \) |
| 71 | \( 1 + (-20.3 - 24.2i)T + (-875. + 4.96e3i)T^{2} \) |
| 73 | \( 1 + (-2.12 - 12.0i)T + (-5.00e3 + 1.82e3i)T^{2} \) |
| 79 | \( 1 + (51.2 - 9.03i)T + (5.86e3 - 2.13e3i)T^{2} \) |
| 83 | \( 1 + (26.2 - 45.4i)T + (-3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + (5.39 + 0.951i)T + (7.44e3 + 2.70e3i)T^{2} \) |
| 97 | \( 1 + (-4.58 + 12.6i)T + (-7.20e3 - 6.04e3i)T^{2} \) |
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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
−15.44922054867893190670407599167, −14.14858772807614758541383677853, −13.03444057027277854259129856809, −12.17238562264433929163302949846, −9.999732492149345477861806750787, −8.738556964046393279743995856010, −8.422158036617557197992073967192, −6.27027308879680087606627968060, −5.59949311099713699911390287109, −2.60156794984316508011045666982,
2.07641650478824426491577082862, 3.49979367655250297585731624411, 6.19793934461318644291269937379, 7.40380885244010989467822841605, 9.710876860935810353682203284553, 10.02353444302320323248103764106, 10.81740677746594617886360204466, 12.61034966635323800225997578885, 13.64318187648944154453238482515, 14.52869913636313564513505626068
