| L(s) = 1 | + (−0.909 − 1.08i)2-s + (0.836 − 2.29i)3-s + (−0.347 + 1.96i)4-s + (−0.852 − 4.83i)5-s + (−3.25 + 1.18i)6-s + (0.583 − 1.00i)7-s + (2.44 − 1.41i)8-s + (2.31 + 1.94i)9-s + (−4.46 + 5.31i)10-s + (1.75 + 3.04i)11-s + (4.23 + 2.44i)12-s + (7.84 + 21.5i)13-s + (−1.62 + 0.286i)14-s + (−11.8 − 2.08i)15-s + (−3.75 − 1.36i)16-s + (7.26 − 6.09i)17-s + ⋯ |
| L(s) = 1 | + (−0.454 − 0.541i)2-s + (0.278 − 0.766i)3-s + (−0.0868 + 0.492i)4-s + (−0.170 − 0.967i)5-s + (−0.541 + 0.197i)6-s + (0.0833 − 0.144i)7-s + (0.306 − 0.176i)8-s + (0.256 + 0.215i)9-s + (−0.446 + 0.531i)10-s + (0.159 + 0.276i)11-s + (0.353 + 0.203i)12-s + (0.603 + 1.65i)13-s + (−0.116 + 0.0204i)14-s + (−0.788 − 0.139i)15-s + (−0.234 − 0.0855i)16-s + (0.427 − 0.358i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0842 + 0.996i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0842 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.677296 - 0.622460i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.677296 - 0.622460i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.909 + 1.08i)T \) |
| 19 | \( 1 + (18.9 - 1.61i)T \) |
| good | 3 | \( 1 + (-0.836 + 2.29i)T + (-6.89 - 5.78i)T^{2} \) |
| 5 | \( 1 + (0.852 + 4.83i)T + (-23.4 + 8.55i)T^{2} \) |
| 7 | \( 1 + (-0.583 + 1.00i)T + (-24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (-1.75 - 3.04i)T + (-60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (-7.84 - 21.5i)T + (-129. + 108. i)T^{2} \) |
| 17 | \( 1 + (-7.26 + 6.09i)T + (50.1 - 284. i)T^{2} \) |
| 23 | \( 1 + (-3.50 + 19.8i)T + (-497. - 180. i)T^{2} \) |
| 29 | \( 1 + (25.7 - 30.6i)T + (-146. - 828. i)T^{2} \) |
| 31 | \( 1 + (43.3 + 25.0i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 - 47.7iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (-7.79 + 21.4i)T + (-1.28e3 - 1.08e3i)T^{2} \) |
| 43 | \( 1 + (-5.98 - 33.9i)T + (-1.73e3 + 632. i)T^{2} \) |
| 47 | \( 1 + (15.8 + 13.2i)T + (383. + 2.17e3i)T^{2} \) |
| 53 | \( 1 + (47.4 + 8.36i)T + (2.63e3 + 960. i)T^{2} \) |
| 59 | \( 1 + (-24.7 - 29.4i)T + (-604. + 3.42e3i)T^{2} \) |
| 61 | \( 1 + (-20.1 + 114. i)T + (-3.49e3 - 1.27e3i)T^{2} \) |
| 67 | \( 1 + (-22.4 + 26.7i)T + (-779. - 4.42e3i)T^{2} \) |
| 71 | \( 1 + (122. - 21.5i)T + (4.73e3 - 1.72e3i)T^{2} \) |
| 73 | \( 1 + (-31.3 - 11.4i)T + (4.08e3 + 3.42e3i)T^{2} \) |
| 79 | \( 1 + (10.2 - 28.0i)T + (-4.78e3 - 4.01e3i)T^{2} \) |
| 83 | \( 1 + (47.2 - 81.9i)T + (-3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + (-3.34 - 9.19i)T + (-6.06e3 + 5.09e3i)T^{2} \) |
| 97 | \( 1 + (6.68 + 7.96i)T + (-1.63e3 + 9.26e3i)T^{2} \) |
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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
−16.28099003313891711251391060438, −14.35468177650179742549814258274, −13.09988378054764320980804386936, −12.35703086037206851888634214520, −11.04275256576520947422333797782, −9.332855069894370833140162916458, −8.313722227097168044175576100044, −6.91056115121550990830654731380, −4.40250977066359909170573848094, −1.66463953694779633362029664467,
3.56466530895799088084398527047, 5.78725759543480327015381328728, 7.40167710606866133197973420579, 8.813510889937428528377349581688, 10.20100375625711784155735437459, 10.98832777694008104525914129194, 12.97768986792171169298556411070, 14.66976539798393974796079824050, 15.14584409038910120497870007712, 16.11764578918337264675914998419
