L(s) = 1 | + (0.951 − 0.690i)2-s + (1.34 − 1.08i)3-s + (−0.190 + 0.587i)4-s + (−0.224 + 0.309i)5-s + (0.530 − 1.96i)6-s + (2.92 + 0.951i)7-s + (0.951 + 2.92i)8-s + (0.633 − 2.93i)9-s + 0.449i·10-s + (0.381 + 1.00i)12-s + (−0.427 − 0.587i)13-s + (3.44 − 1.11i)14-s + (0.0335 + 0.660i)15-s + (1.92 + 1.40i)16-s + (−3.44 − 2.5i)17-s + (−1.42 − 3.22i)18-s + ⋯ |
L(s) = 1 | + (0.672 − 0.488i)2-s + (0.778 − 0.628i)3-s + (−0.0954 + 0.293i)4-s + (−0.100 + 0.138i)5-s + (0.216 − 0.802i)6-s + (1.10 + 0.359i)7-s + (0.336 + 1.03i)8-s + (0.211 − 0.977i)9-s + 0.141i·10-s + (0.110 + 0.288i)12-s + (−0.118 − 0.163i)13-s + (0.919 − 0.298i)14-s + (0.00865 + 0.170i)15-s + (0.481 + 0.350i)16-s + (−0.834 − 0.606i)17-s + (−0.335 − 0.760i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.824 + 0.565i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.824 + 0.565i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.31102 - 0.716642i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.31102 - 0.716642i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-1.34 + 1.08i)T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-0.951 + 0.690i)T + (0.618 - 1.90i)T^{2} \) |
| 5 | \( 1 + (0.224 - 0.309i)T + (-1.54 - 4.75i)T^{2} \) |
| 7 | \( 1 + (-2.92 - 0.951i)T + (5.66 + 4.11i)T^{2} \) |
| 13 | \( 1 + (0.427 + 0.587i)T + (-4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (3.44 + 2.5i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (2.5 - 0.812i)T + (15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + 6.23iT - 23T^{2} \) |
| 29 | \( 1 + (-0.726 + 2.23i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (4.73 - 3.44i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (-0.927 + 2.85i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (0.224 + 0.690i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 - 6.88iT - 43T^{2} \) |
| 47 | \( 1 + (7.91 - 2.57i)T + (38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (-1.08 - 1.5i)T + (-16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (0.363 + 0.118i)T + (47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (1.54 - 2.12i)T + (-18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 - 7.32T + 67T^{2} \) |
| 71 | \( 1 + (-3.13 + 4.30i)T + (-21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-8.94 - 2.90i)T + (59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (2.33 + 3.21i)T + (-24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (-5.11 - 3.71i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 - 0.527iT - 89T^{2} \) |
| 97 | \( 1 + (-3.54 + 2.57i)T + (29.9 - 92.2i)T^{2} \) |
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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
−11.50014165096812903616873967360, −10.83789531685845365022118039724, −9.241468962057262035623451538866, −8.414262609326560394524184310218, −7.78455183228575317572713061131, −6.66537896243804598870649534505, −5.13161853290926275920356755934, −4.16123765551905636689403451355, −2.88368423328129177694388737918, −1.92780243362856542071915554372,
1.86375941074100668139664337168, 3.77060262560312772726681253761, 4.55789483018797076670353202985, 5.33199859997694958943109284712, 6.71645199445842195595180943389, 7.79757644709678873047554367899, 8.665274994073799301627719589693, 9.657690836494184213005156823090, 10.58965842834918481405516958301, 11.31577704344352975081871177225