| 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
−14.52869913636313564513505626068, −13.64318187648944154453238482515, −12.61034966635323800225997578885, −10.81740677746594617886360204466, −10.02353444302320323248103764106, −9.710876860935810353682203284553, −7.40380885244010989467822841605, −6.19793934461318644291269937379, −3.49979367655250297585731624411, −2.07641650478824426491577082862,
2.60156794984316508011045666982, 5.59949311099713699911390287109, 6.27027308879680087606627968060, 8.422158036617557197992073967192, 8.738556964046393279743995856010, 9.999732492149345477861806750787, 12.17238562264433929163302949846, 13.03444057027277854259129856809, 14.14858772807614758541383677853, 15.44922054867893190670407599167
