| L(s) = 1 | + (−0.626 + 0.174i)2-s + (−1.18 − 1.26i)3-s + (−1.34 + 0.814i)4-s + (0.644 + 0.0250i)5-s + (0.962 + 0.586i)6-s + (−0.504 + 0.0789i)7-s + (1.59 − 1.69i)8-s + (−0.202 + 2.99i)9-s + (−0.408 + 0.0967i)10-s + (−0.486 + 3.56i)11-s + (2.62 + 0.744i)12-s + (0.0982 + 0.143i)13-s + (0.302 − 0.137i)14-s + (−0.730 − 0.844i)15-s + (0.763 − 1.44i)16-s + (6.29 + 3.16i)17-s + ⋯ |
| L(s) = 1 | + (−0.443 + 0.123i)2-s + (−0.682 − 0.730i)3-s + (−0.674 + 0.407i)4-s + (0.288 + 0.0111i)5-s + (0.392 + 0.239i)6-s + (−0.190 + 0.0298i)7-s + (0.564 − 0.598i)8-s + (−0.0673 + 0.997i)9-s + (−0.129 + 0.0305i)10-s + (−0.146 + 1.07i)11-s + (0.758 + 0.214i)12-s + (0.0272 + 0.0397i)13-s + (0.0808 − 0.0367i)14-s + (−0.188 − 0.218i)15-s + (0.190 − 0.362i)16-s + (1.52 + 0.766i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.387 - 0.921i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.387 - 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.496791 + 0.330008i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.496791 + 0.330008i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.18 + 1.26i)T \) |
| good | 2 | \( 1 + (0.626 - 0.174i)T + (1.71 - 1.03i)T^{2} \) |
| 5 | \( 1 + (-0.644 - 0.0250i)T + (4.98 + 0.387i)T^{2} \) |
| 7 | \( 1 + (0.504 - 0.0789i)T + (6.66 - 2.13i)T^{2} \) |
| 11 | \( 1 + (0.486 - 3.56i)T + (-10.5 - 2.94i)T^{2} \) |
| 13 | \( 1 + (-0.0982 - 0.143i)T + (-4.68 + 12.1i)T^{2} \) |
| 17 | \( 1 + (-6.29 - 3.16i)T + (10.1 + 13.6i)T^{2} \) |
| 19 | \( 1 + (0.433 - 7.43i)T + (-18.8 - 2.20i)T^{2} \) |
| 23 | \( 1 + (-2.32 - 6.02i)T + (-17.0 + 15.4i)T^{2} \) |
| 29 | \( 1 + (-3.25 + 2.32i)T + (9.38 - 27.4i)T^{2} \) |
| 31 | \( 1 + (5.44 + 6.23i)T + (-4.19 + 30.7i)T^{2} \) |
| 37 | \( 1 + (-3.71 + 4.98i)T + (-10.6 - 35.4i)T^{2} \) |
| 41 | \( 1 + (-0.706 - 2.74i)T + (-35.8 + 19.8i)T^{2} \) |
| 43 | \( 1 + (5.41 - 4.90i)T + (4.16 - 42.7i)T^{2} \) |
| 47 | \( 1 + (5.10 - 5.84i)T + (-6.36 - 46.5i)T^{2} \) |
| 53 | \( 1 + (1.26 + 7.14i)T + (-49.8 + 18.1i)T^{2} \) |
| 59 | \( 1 + (-0.549 - 0.224i)T + (42.1 + 41.3i)T^{2} \) |
| 61 | \( 1 + (-3.13 - 1.89i)T + (28.4 + 53.9i)T^{2} \) |
| 67 | \( 1 + (-5.50 - 3.93i)T + (21.6 + 63.3i)T^{2} \) |
| 71 | \( 1 + (-3.45 + 11.5i)T + (-59.3 - 39.0i)T^{2} \) |
| 73 | \( 1 + (-8.73 - 2.06i)T + (65.2 + 32.7i)T^{2} \) |
| 79 | \( 1 + (-0.340 - 0.333i)T + (1.53 + 78.9i)T^{2} \) |
| 83 | \( 1 + (0.138 - 0.536i)T + (-72.6 - 40.0i)T^{2} \) |
| 89 | \( 1 + (-3.77 - 12.6i)T + (-74.3 + 48.9i)T^{2} \) |
| 97 | \( 1 + (6.14 - 0.238i)T + (96.7 - 7.51i)T^{2} \) |
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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
−12.55071391387707600676197241117, −11.46608787997934516972463827397, −10.02522242539867095139066881361, −9.678803165667081487982160293320, −7.891733053495590867109055930649, −7.73683434933630886404416392089, −6.24012408741794452551556647116, −5.24334493275465664299932984097, −3.78446443572824445020797451564, −1.60590934447590459402048216421,
0.66326020905576188804167024166, 3.27177693350421220383411254229, 4.84748136584629771110733104273, 5.50394177859043190723522540983, 6.74197823270075815003253615505, 8.394319955602420125419420012226, 9.215306814492580524856228679220, 10.03120290922957381972354904276, 10.78130942442016352442496133299, 11.61082198580361921645260821742
