| L(s) = 1 | + (−0.939 − 0.342i)2-s + (−1.14 − 1.29i)3-s + (0.766 + 0.642i)4-s + (0.617 − 3.49i)5-s + (0.631 + 1.61i)6-s + (−0.244 + 0.205i)7-s + (−0.500 − 0.866i)8-s + (−0.377 + 2.97i)9-s + (−1.77 + 3.07i)10-s + (0.773 + 4.38i)11-s + (−0.0419 − 1.73i)12-s + (4.39 − 1.60i)13-s + (0.300 − 0.109i)14-s + (−5.25 + 3.20i)15-s + (0.173 + 0.984i)16-s + (−0.567 + 0.982i)17-s + ⋯ |
| L(s) = 1 | + (−0.664 − 0.241i)2-s + (−0.661 − 0.750i)3-s + (0.383 + 0.321i)4-s + (0.275 − 1.56i)5-s + (0.257 + 0.658i)6-s + (−0.0925 + 0.0776i)7-s + (−0.176 − 0.306i)8-s + (−0.125 + 0.992i)9-s + (−0.561 + 0.973i)10-s + (0.233 + 1.32i)11-s + (−0.0121 − 0.499i)12-s + (1.21 − 0.443i)13-s + (0.0802 − 0.0292i)14-s + (−1.35 + 0.827i)15-s + (0.0434 + 0.246i)16-s + (−0.137 + 0.238i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.130 + 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.130 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.427387 - 0.374810i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.427387 - 0.374810i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.939 + 0.342i)T \) |
| 3 | \( 1 + (1.14 + 1.29i)T \) |
| good | 5 | \( 1 + (-0.617 + 3.49i)T + (-4.69 - 1.71i)T^{2} \) |
| 7 | \( 1 + (0.244 - 0.205i)T + (1.21 - 6.89i)T^{2} \) |
| 11 | \( 1 + (-0.773 - 4.38i)T + (-10.3 + 3.76i)T^{2} \) |
| 13 | \( 1 + (-4.39 + 1.60i)T + (9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (0.567 - 0.982i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.928 + 1.60i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.110 - 0.0926i)T + (3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (-4.09 - 1.49i)T + (22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (-0.514 - 0.431i)T + (5.38 + 30.5i)T^{2} \) |
| 37 | \( 1 + (3.79 - 6.57i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (2.04 - 0.744i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (1.23 + 6.98i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (7.91 - 6.63i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 - 0.805T + 53T^{2} \) |
| 59 | \( 1 + (-0.517 + 2.93i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (2.67 - 2.24i)T + (10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (6.99 - 2.54i)T + (51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (-4.04 + 7.01i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (7.30 + 12.6i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-11.8 - 4.30i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (-5.08 - 1.85i)T + (63.5 + 53.3i)T^{2} \) |
| 89 | \( 1 + (2.52 + 4.37i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-3.24 - 18.3i)T + (-91.1 + 33.1i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.56832045060345316976878043247, −13.47216914380084608913108096772, −12.66383286458399918696398095680, −11.90260081117201195592799047914, −10.45635523574424483948992755587, −9.060438024757833575397109753500, −8.018126147281152179353968330278, −6.43006674570982844850435019174, −4.86079399754260743684800646591, −1.47095584505163990617267129332,
3.45391049160437654506309570118, 5.95663554087565013889154636311, 6.71241493544558731484101917092, 8.630142869683277440539490204559, 10.01504074947853443799202340980, 10.91256518198200206661807125523, 11.52262382244825643286417455686, 13.77457170470180612909831972966, 14.73041889051877472384383099335, 15.85307927996238967404995827641
