| 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
−16.11764578918337264675914998419, −15.14584409038910120497870007712, −14.66976539798393974796079824050, −12.97768986792171169298556411070, −10.98832777694008104525914129194, −10.20100375625711784155735437459, −8.813510889937428528377349581688, −7.40167710606866133197973420579, −5.78725759543480327015381328728, −3.56466530895799088084398527047,
1.66463953694779633362029664467, 4.40250977066359909170573848094, 6.91056115121550990830654731380, 8.313722227097168044175576100044, 9.332855069894370833140162916458, 11.04275256576520947422333797782, 12.35703086037206851888634214520, 13.09988378054764320980804386936, 14.35468177650179742549814258274, 16.28099003313891711251391060438
