| L(s) = 1 | + (−0.965 − 0.258i)2-s + (−0.540 + 1.64i)3-s + (0.866 + 0.499i)4-s + (−1.97 − 0.529i)5-s + (0.947 − 1.44i)6-s + (2.60 + 0.450i)7-s + (−0.707 − 0.707i)8-s + (−2.41 − 1.77i)9-s + (1.77 + 1.02i)10-s + (−4.21 + 1.12i)11-s + (−1.29 + 1.15i)12-s + (−1.63 − 3.21i)13-s + (−2.40 − 1.11i)14-s + (1.93 − 2.96i)15-s + (0.500 + 0.866i)16-s + (1.80 − 3.13i)17-s + ⋯ |
| L(s) = 1 | + (−0.683 − 0.183i)2-s + (−0.311 + 0.950i)3-s + (0.433 + 0.249i)4-s + (−0.883 − 0.236i)5-s + (0.386 − 0.591i)6-s + (0.985 + 0.170i)7-s + (−0.249 − 0.249i)8-s + (−0.805 − 0.592i)9-s + (0.560 + 0.323i)10-s + (−1.27 + 0.340i)11-s + (−0.372 + 0.333i)12-s + (−0.454 − 0.890i)13-s + (−0.641 − 0.296i)14-s + (0.500 − 0.765i)15-s + (0.125 + 0.216i)16-s + (0.438 − 0.759i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.112 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.112 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.243708 - 0.272823i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.243708 - 0.272823i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.965 + 0.258i)T \) |
| 3 | \( 1 + (0.540 - 1.64i)T \) |
| 7 | \( 1 + (-2.60 - 0.450i)T \) |
| 13 | \( 1 + (1.63 + 3.21i)T \) |
| good | 5 | \( 1 + (1.97 + 0.529i)T + (4.33 + 2.5i)T^{2} \) |
| 11 | \( 1 + (4.21 - 1.12i)T + (9.52 - 5.5i)T^{2} \) |
| 17 | \( 1 + (-1.80 + 3.13i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.449 - 1.67i)T + (-16.4 - 9.5i)T^{2} \) |
| 23 | \( 1 + (-0.976 - 1.69i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 7.12iT - 29T^{2} \) |
| 31 | \( 1 + (1.91 - 0.513i)T + (26.8 - 15.5i)T^{2} \) |
| 37 | \( 1 + (2.87 + 0.771i)T + (32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (-5.01 + 5.01i)T - 41iT^{2} \) |
| 43 | \( 1 + 12.4iT - 43T^{2} \) |
| 47 | \( 1 + (0.536 - 2.00i)T + (-40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (6.45 + 3.72i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-11.3 + 3.03i)T + (51.0 - 29.5i)T^{2} \) |
| 61 | \( 1 + (5.54 + 9.60i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (7.55 - 2.02i)T + (58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + (5.58 - 5.58i)T - 71iT^{2} \) |
| 73 | \( 1 + (-1.97 - 7.38i)T + (-63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (4.09 + 7.09i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (6.47 - 6.47i)T - 83iT^{2} \) |
| 89 | \( 1 + (-0.741 + 2.76i)T + (-77.0 - 44.5i)T^{2} \) |
| 97 | \( 1 + (0.184 + 0.184i)T + 97iT^{2} \) |
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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
−10.47250189793610640100550875010, −9.893822945418822372540020542602, −8.773491526830524173435142314008, −7.950698387518082538407084887502, −7.44335147580596910889945606329, −5.62399917920651953630036087342, −4.95562809124145413017601270129, −3.81326502961133660054403954702, −2.51163854870208096686520505782, −0.27679216946873281049507045470,
1.48849328508875195683017476704, 2.84696235773405746737187342359, 4.60691363535000777976938736682, 5.66140206549300146449372285275, 6.84155816061138778230650287579, 7.65475250211535932785657452171, 8.046284882213583957842043542518, 8.961999431330041847895042912094, 10.46207351150358332259979634386, 11.05979653748936635391207756924
