| L(s) = 1 | + (0.138 − 1.40i)2-s + (0.123 + 1.72i)3-s + (−1.96 − 0.389i)4-s + (0.608 + 1.67i)5-s + (2.44 + 0.0650i)6-s + (0.408 + 2.31i)7-s + (−0.820 + 2.70i)8-s + (−2.96 + 0.427i)9-s + (2.43 − 0.624i)10-s + (−0.860 + 2.36i)11-s + (0.430 − 3.43i)12-s + (0.359 − 0.428i)13-s + (3.31 − 0.253i)14-s + (−2.81 + 1.25i)15-s + (3.69 + 1.52i)16-s + (1.49 − 2.59i)17-s + ⋯ |
| L(s) = 1 | + (0.0979 − 0.995i)2-s + (0.0714 + 0.997i)3-s + (−0.980 − 0.194i)4-s + (0.272 + 0.747i)5-s + (0.999 + 0.0265i)6-s + (0.154 + 0.874i)7-s + (−0.290 + 0.956i)8-s + (−0.989 + 0.142i)9-s + (0.770 − 0.197i)10-s + (−0.259 + 0.712i)11-s + (0.124 − 0.992i)12-s + (0.0997 − 0.118i)13-s + (0.885 − 0.0678i)14-s + (−0.726 + 0.324i)15-s + (0.923 + 0.382i)16-s + (0.363 − 0.629i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.779 - 0.626i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.779 - 0.626i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.09457 + 0.385098i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.09457 + 0.385098i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.138 + 1.40i)T \) |
| 3 | \( 1 + (-0.123 - 1.72i)T \) |
| good | 5 | \( 1 + (-0.608 - 1.67i)T + (-3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (-0.408 - 2.31i)T + (-6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (0.860 - 2.36i)T + (-8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (-0.359 + 0.428i)T + (-2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-1.49 + 2.59i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-6.64 + 3.83i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.12 - 6.39i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (5.22 + 6.22i)T + (-5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (0.643 - 3.65i)T + (-29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (-4.07 - 2.35i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (5.67 + 4.76i)T + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-0.928 + 2.55i)T + (-32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (-0.0919 - 0.521i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 - 6.39iT - 53T^{2} \) |
| 59 | \( 1 + (-2.87 - 7.88i)T + (-45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-0.716 + 0.126i)T + (57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-3.68 + 4.39i)T + (-11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-7.38 + 12.7i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (5.98 + 10.3i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (8.94 - 7.50i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-4.81 - 5.74i)T + (-14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (1.16 + 2.02i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-13.8 - 5.05i)T + (74.3 + 62.3i)T^{2} \) |
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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
−11.98528437019827045075851970104, −11.48798610921593829077598604273, −10.45088635614150918151919983658, −9.641874379496949121207695205073, −9.047430113927296816485401517185, −7.60947286565531893716168935826, −5.70085166941663863212402978890, −4.92488490860444511109798775528, −3.41245322300996201023027012124, −2.44175501300399753166966136090,
1.05793602272321552868142306156, 3.59158066157805946643572399216, 5.18089094724673904215697798013, 6.07198221399098633426325738911, 7.23285686505059406206987623103, 8.047182586522184379265671641166, 8.827732254674152631145128607266, 10.04992935431361773454003222112, 11.45867358920185702047134390796, 12.76922040374830635653016587072
