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
−11.20745401803415298972313401754, −10.63601839856690942740460250320, −9.658187309006631975401592354573, −8.486978857448767431842583649262, −8.135032817207738478902669974591, −7.23124520677279585511289260645, −5.84988675062913517257130757471, −4.96941937060700016714958120504, −3.20875018702129280922203278434, −1.80799393106828595879054532892,
0.976241974130237568034308365490, 2.38332550002166264323397393711, 3.87356879346951166266442580441, 4.93880750104129598131629847120, 6.91381569241330854862623155774, 7.58767183445113198326540436727, 8.444503642325927655471227552612, 9.007523569424575924303754524835, 10.31321380904582214750787083545, 10.97351547348888139644178478760