| L(s) = 1 | + (−2.44 − 1.41i)2-s + (8.60 + 2.62i)3-s + (3.99 + 6.92i)4-s + (3.82 + 2.20i)5-s + (−17.3 − 18.6i)6-s + (2.10 + 48.9i)7-s − 22.6i·8-s + (67.2 + 45.1i)9-s + (−6.24 − 10.8i)10-s + (81.1 − 46.8i)11-s + (16.2 + 70.1i)12-s + 275.·13-s + (64.0 − 122. i)14-s + (27.1 + 29.0i)15-s + (−32.0 + 55.4i)16-s + (−296. + 171. i)17-s + ⋯ |
| L(s) = 1 | + (−0.612 − 0.353i)2-s + (0.956 + 0.291i)3-s + (0.249 + 0.433i)4-s + (0.152 + 0.0883i)5-s + (−0.482 − 0.516i)6-s + (0.0430 + 0.999i)7-s − 0.353i·8-s + (0.829 + 0.557i)9-s + (−0.0624 − 0.108i)10-s + (0.670 − 0.387i)11-s + (0.112 + 0.487i)12-s + 1.63·13-s + (0.326 − 0.627i)14-s + (0.120 + 0.129i)15-s + (−0.125 + 0.216i)16-s + (−1.02 + 0.592i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 - 0.327i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.944 - 0.327i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.56261 + 0.263470i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.56261 + 0.263470i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (2.44 + 1.41i)T \) |
| 3 | \( 1 + (-8.60 - 2.62i)T \) |
| 7 | \( 1 + (-2.10 - 48.9i)T \) |
| good | 5 | \( 1 + (-3.82 - 2.20i)T + (312.5 + 541. i)T^{2} \) |
| 11 | \( 1 + (-81.1 + 46.8i)T + (7.32e3 - 1.26e4i)T^{2} \) |
| 13 | \( 1 - 275.T + 2.85e4T^{2} \) |
| 17 | \( 1 + (296. - 171. i)T + (4.17e4 - 7.23e4i)T^{2} \) |
| 19 | \( 1 + (-34.1 + 59.1i)T + (-6.51e4 - 1.12e5i)T^{2} \) |
| 23 | \( 1 + (675. + 389. i)T + (1.39e5 + 2.42e5i)T^{2} \) |
| 29 | \( 1 - 300. iT - 7.07e5T^{2} \) |
| 31 | \( 1 + (136. + 235. i)T + (-4.61e5 + 7.99e5i)T^{2} \) |
| 37 | \( 1 + (19.9 - 34.4i)T + (-9.37e5 - 1.62e6i)T^{2} \) |
| 41 | \( 1 + 1.68e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 + 1.31e3T + 3.41e6T^{2} \) |
| 47 | \( 1 + (742. + 428. i)T + (2.43e6 + 4.22e6i)T^{2} \) |
| 53 | \( 1 + (-3.78e3 + 2.18e3i)T + (3.94e6 - 6.83e6i)T^{2} \) |
| 59 | \( 1 + (1.57e3 - 908. i)T + (6.05e6 - 1.04e7i)T^{2} \) |
| 61 | \( 1 + (-2.64e3 + 4.57e3i)T + (-6.92e6 - 1.19e7i)T^{2} \) |
| 67 | \( 1 + (-2.26e3 - 3.92e3i)T + (-1.00e7 + 1.74e7i)T^{2} \) |
| 71 | \( 1 - 7.78e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + (4.68e3 + 8.11e3i)T + (-1.41e7 + 2.45e7i)T^{2} \) |
| 79 | \( 1 + (-1.83e3 + 3.18e3i)T + (-1.94e7 - 3.37e7i)T^{2} \) |
| 83 | \( 1 - 943. iT - 4.74e7T^{2} \) |
| 89 | \( 1 + (5.15e3 + 2.97e3i)T + (3.13e7 + 5.43e7i)T^{2} \) |
| 97 | \( 1 - 1.11e4T + 8.85e7T^{2} \) |
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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
−15.49050216210788030133361821508, −14.18459235803379642155013114081, −13.02762072613836246572074766418, −11.56913710353781997843402956205, −10.28594643456809392607787041064, −8.878661640705768899505189442036, −8.387370861047265723508189786998, −6.32632996522475872029754697545, −3.81286183004640381531924215830, −2.05627121034309968361500202373,
1.45767512496323872544643271336, 3.91082883552404324920466337185, 6.47671474200809131824998977064, 7.66885258099286437188211972324, 8.883301357913018499201710797151, 9.966341980132785008612674200326, 11.43463496010845338570115390757, 13.33521689626024229739137889443, 13.95336253508847761597111252786, 15.28657771476582494956780045551
