| L(s) = 1 | + (−0.342 + 0.939i)2-s + (−1.68 − 0.395i)3-s + (−0.766 − 0.642i)4-s + (−0.330 + 1.87i)5-s + (0.948 − 1.44i)6-s + (2.54 − 0.716i)7-s + (0.866 − 0.500i)8-s + (2.68 + 1.33i)9-s + (−1.64 − 0.951i)10-s + (−1.99 + 0.352i)11-s + (1.03 + 1.38i)12-s + (0.169 + 0.466i)13-s + (−0.197 + 2.63i)14-s + (1.29 − 3.02i)15-s + (0.173 + 0.984i)16-s + (−2.21 + 3.83i)17-s + ⋯ |
| L(s) = 1 | + (−0.241 + 0.664i)2-s + (−0.973 − 0.228i)3-s + (−0.383 − 0.321i)4-s + (−0.147 + 0.838i)5-s + (0.387 − 0.591i)6-s + (0.962 − 0.270i)7-s + (0.306 − 0.176i)8-s + (0.895 + 0.444i)9-s + (−0.521 − 0.300i)10-s + (−0.602 + 0.106i)11-s + (0.299 + 0.400i)12-s + (0.0470 + 0.129i)13-s + (−0.0527 + 0.705i)14-s + (0.335 − 0.782i)15-s + (0.0434 + 0.246i)16-s + (−0.537 + 0.930i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.760 - 0.649i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.760 - 0.649i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.224461 + 0.608150i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.224461 + 0.608150i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.342 - 0.939i)T \) |
| 3 | \( 1 + (1.68 + 0.395i)T \) |
| 7 | \( 1 + (-2.54 + 0.716i)T \) |
| good | 5 | \( 1 + (0.330 - 1.87i)T + (-4.69 - 1.71i)T^{2} \) |
| 11 | \( 1 + (1.99 - 0.352i)T + (10.3 - 3.76i)T^{2} \) |
| 13 | \( 1 + (-0.169 - 0.466i)T + (-9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (2.21 - 3.83i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (3.77 - 2.18i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (3.43 - 4.09i)T + (-3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (2.07 - 5.69i)T + (-22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (0.836 - 0.997i)T + (-5.38 - 30.5i)T^{2} \) |
| 37 | \( 1 + (3.22 - 5.58i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.67 + 0.609i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (0.194 + 1.10i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (-8.86 + 7.43i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 - 6.77iT - 53T^{2} \) |
| 59 | \( 1 + (0.400 - 2.27i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (3.31 + 3.95i)T + (-10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (-9.96 + 3.62i)T + (51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (0.688 + 0.397i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (11.1 - 6.43i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (13.3 + 4.86i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (-10.7 - 3.90i)T + (63.5 + 53.3i)T^{2} \) |
| 89 | \( 1 + (-1.09 - 1.89i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-11.7 + 2.06i)T + (91.1 - 33.1i)T^{2} \) |
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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
−11.50589620920757858276388401310, −10.62664001436736872836679456098, −10.33415011838807583248479246845, −8.716064374869203399534765575461, −7.71670272049802728253271359521, −7.03064368762445878048304069417, −6.07513190088577353605019029273, −5.11858353023524643500300181257, −3.99539795922888528579159731947, −1.77478961191863922531558555790,
0.53825884226678132177318742827, 2.23722025493231352951810444033, 4.28706080998033947921660107743, 4.85386898581842320023739586963, 5.90685350608727664718519070973, 7.38837695747230781746772918045, 8.444026916578378712453360164823, 9.223122851526635222337809366075, 10.36388938403760809924118254556, 11.08024950362400372293727160429
