| L(s) = 1 | + (−1.23 − 0.689i)2-s + (0.881 − 0.471i)3-s + (1.04 + 1.70i)4-s + (−0.913 − 1.11i)5-s + (−1.41 − 0.0260i)6-s + (3.00 − 2.01i)7-s + (−0.121 − 2.82i)8-s + (0.555 − 0.831i)9-s + (0.360 + 2.00i)10-s + (−2.39 + 0.725i)11-s + (1.72 + 1.00i)12-s + (1.40 + 1.15i)13-s + (−5.10 + 0.407i)14-s + (−1.33 − 0.551i)15-s + (−1.79 + 3.57i)16-s + (0.984 − 0.407i)17-s + ⋯ |
| L(s) = 1 | + (−0.873 − 0.487i)2-s + (0.509 − 0.272i)3-s + (0.524 + 0.851i)4-s + (−0.408 − 0.497i)5-s + (−0.577 − 0.0106i)6-s + (1.13 − 0.759i)7-s + (−0.0429 − 0.999i)8-s + (0.185 − 0.277i)9-s + (0.114 + 0.633i)10-s + (−0.721 + 0.218i)11-s + (0.498 + 0.290i)12-s + (0.390 + 0.320i)13-s + (−1.36 + 0.109i)14-s + (−0.343 − 0.142i)15-s + (−0.449 + 0.893i)16-s + (0.238 − 0.0989i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0796 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0796 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.730198 - 0.790900i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.730198 - 0.790900i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.23 + 0.689i)T \) |
| 3 | \( 1 + (-0.881 + 0.471i)T \) |
| good | 5 | \( 1 + (0.913 + 1.11i)T + (-0.975 + 4.90i)T^{2} \) |
| 7 | \( 1 + (-3.00 + 2.01i)T + (2.67 - 6.46i)T^{2} \) |
| 11 | \( 1 + (2.39 - 0.725i)T + (9.14 - 6.11i)T^{2} \) |
| 13 | \( 1 + (-1.40 - 1.15i)T + (2.53 + 12.7i)T^{2} \) |
| 17 | \( 1 + (-0.984 + 0.407i)T + (12.0 - 12.0i)T^{2} \) |
| 19 | \( 1 + (0.506 + 5.14i)T + (-18.6 + 3.70i)T^{2} \) |
| 23 | \( 1 + (-3.05 + 0.608i)T + (21.2 - 8.80i)T^{2} \) |
| 29 | \( 1 + (0.237 - 0.782i)T + (-24.1 - 16.1i)T^{2} \) |
| 31 | \( 1 + (-0.899 - 0.899i)T + 31iT^{2} \) |
| 37 | \( 1 + (8.18 + 0.806i)T + (36.2 + 7.21i)T^{2} \) |
| 41 | \( 1 + (2.17 + 10.9i)T + (-37.8 + 15.6i)T^{2} \) |
| 43 | \( 1 + (-2.89 - 1.54i)T + (23.8 + 35.7i)T^{2} \) |
| 47 | \( 1 + (-2.42 - 5.85i)T + (-33.2 + 33.2i)T^{2} \) |
| 53 | \( 1 + (0.598 + 1.97i)T + (-44.0 + 29.4i)T^{2} \) |
| 59 | \( 1 + (-2.14 + 1.76i)T + (11.5 - 57.8i)T^{2} \) |
| 61 | \( 1 + (-2.46 - 4.60i)T + (-33.8 + 50.7i)T^{2} \) |
| 67 | \( 1 + (-6.78 - 12.6i)T + (-37.2 + 55.7i)T^{2} \) |
| 71 | \( 1 + (-5.12 - 7.66i)T + (-27.1 + 65.5i)T^{2} \) |
| 73 | \( 1 + (8.14 + 5.44i)T + (27.9 + 67.4i)T^{2} \) |
| 79 | \( 1 + (2.22 - 5.36i)T + (-55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + (-3.06 + 0.301i)T + (81.4 - 16.1i)T^{2} \) |
| 89 | \( 1 + (-14.1 - 2.81i)T + (82.2 + 34.0i)T^{2} \) |
| 97 | \( 1 + (6.26 + 6.26i)T + 97iT^{2} \) |
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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
−10.97351547348888139644178478760, −10.31321380904582214750787083545, −9.007523569424575924303754524835, −8.444503642325927655471227552612, −7.58767183445113198326540436727, −6.91381569241330854862623155774, −4.93880750104129598131629847120, −3.87356879346951166266442580441, −2.38332550002166264323397393711, −0.976241974130237568034308365490,
1.80799393106828595879054532892, 3.20875018702129280922203278434, 4.96941937060700016714958120504, 5.84988675062913517257130757471, 7.23124520677279585511289260645, 8.135032817207738478902669974591, 8.486978857448767431842583649262, 9.658187309006631975401592354573, 10.63601839856690942740460250320, 11.20745401803415298972313401754
