| L(s) = 1 | + (0.551 − 1.69i)2-s + (0.360 − 1.69i)3-s + (0.662 + 0.481i)4-s + (−3.20 − 5.54i)5-s + (−2.67 − 1.54i)6-s + (−2.32 + 1.03i)7-s + (6.95 − 5.05i)8-s + (−2.74 − 1.22i)9-s + (−11.1 + 2.37i)10-s + (4.64 + 0.487i)11-s + (1.05 − 0.948i)12-s + (2.80 + 2.52i)13-s + (0.475 + 4.51i)14-s + (−10.5 + 3.42i)15-s + (−3.72 − 11.4i)16-s + (−4.72 + 0.496i)17-s + ⋯ |
| L(s) = 1 | + (0.275 − 0.848i)2-s + (0.120 − 0.564i)3-s + (0.165 + 0.120i)4-s + (−0.640 − 1.10i)5-s + (−0.445 − 0.257i)6-s + (−0.332 + 0.148i)7-s + (0.869 − 0.631i)8-s + (−0.304 − 0.135i)9-s + (−1.11 + 0.237i)10-s + (0.422 + 0.0443i)11-s + (0.0877 − 0.0790i)12-s + (0.215 + 0.194i)13-s + (0.0339 + 0.322i)14-s + (−0.703 + 0.228i)15-s + (−0.232 − 0.716i)16-s + (−0.277 + 0.0291i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.425 + 0.904i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.425 + 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.832718 - 1.31229i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.832718 - 1.31229i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.360 + 1.69i)T \) |
| 31 | \( 1 + (-21.3 - 22.4i)T \) |
| good | 2 | \( 1 + (-0.551 + 1.69i)T + (-3.23 - 2.35i)T^{2} \) |
| 5 | \( 1 + (3.20 + 5.54i)T + (-12.5 + 21.6i)T^{2} \) |
| 7 | \( 1 + (2.32 - 1.03i)T + (32.7 - 36.4i)T^{2} \) |
| 11 | \( 1 + (-4.64 - 0.487i)T + (118. + 25.1i)T^{2} \) |
| 13 | \( 1 + (-2.80 - 2.52i)T + (17.6 + 168. i)T^{2} \) |
| 17 | \( 1 + (4.72 - 0.496i)T + (282. - 60.0i)T^{2} \) |
| 19 | \( 1 + (-12.3 - 13.7i)T + (-37.7 + 359. i)T^{2} \) |
| 23 | \( 1 + (-2.29 - 3.15i)T + (-163. + 503. i)T^{2} \) |
| 29 | \( 1 + (-52.2 - 16.9i)T + (680. + 494. i)T^{2} \) |
| 37 | \( 1 + (23.7 + 13.6i)T + (684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + (30.9 - 6.58i)T + (1.53e3 - 683. i)T^{2} \) |
| 43 | \( 1 + (2.67 - 2.40i)T + (193. - 1.83e3i)T^{2} \) |
| 47 | \( 1 + (14.4 + 44.5i)T + (-1.78e3 + 1.29e3i)T^{2} \) |
| 53 | \( 1 + (17.5 - 39.3i)T + (-1.87e3 - 2.08e3i)T^{2} \) |
| 59 | \( 1 + (94.9 + 20.1i)T + (3.18e3 + 1.41e3i)T^{2} \) |
| 61 | \( 1 + 2.35iT - 3.72e3T^{2} \) |
| 67 | \( 1 + (-15.8 - 27.5i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 + (35.3 + 15.7i)T + (3.37e3 + 3.74e3i)T^{2} \) |
| 73 | \( 1 + (122. + 12.8i)T + (5.21e3 + 1.10e3i)T^{2} \) |
| 79 | \( 1 + (-130. + 13.7i)T + (6.10e3 - 1.29e3i)T^{2} \) |
| 83 | \( 1 + (29.7 + 140. i)T + (-6.29e3 + 2.80e3i)T^{2} \) |
| 89 | \( 1 + (46.8 - 64.4i)T + (-2.44e3 - 7.53e3i)T^{2} \) |
| 97 | \( 1 + (-147. - 106. i)T + (2.90e3 + 8.94e3i)T^{2} \) |
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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
−13.11237999819505895182493727002, −12.12869472263719994524042980680, −11.91235705889122350128281167127, −10.42949403572807208885200186006, −8.979098484572061886594415538184, −7.916130787181671177464058709057, −6.60170032130111545491256895123, −4.68652714821294779485823887614, −3.27623713343374066024661614664, −1.33635113490690794951313748129,
3.05351659862283623101384724549, 4.63225068323859790164439684325, 6.26077695799930704093260540518, 7.09084127853525969129304274331, 8.289508378896163970191936051278, 9.934331911147509244875377496320, 10.91056273361236355682157936867, 11.73877140690965948997096298048, 13.60477038058454015266699023315, 14.36281796244657918716889506466
