| L(s) = 1 | + (1.12 − 3.46i)2-s + (4.32 + 14.9i)3-s + (15.1 + 10.9i)4-s + (−54.0 + 17.5i)5-s + (56.8 + 1.88i)6-s + (−123. + 170. i)7-s + (149. − 108. i)8-s + (−205. + 129. i)9-s + 207. i·10-s + (371. − 151. i)11-s + (−99.1 + 274. i)12-s + (787. + 255. i)13-s + (450. + 620. i)14-s + (−496. − 733. i)15-s + (−23.4 − 72.1i)16-s + (−271. − 836. i)17-s + ⋯ |
| L(s) = 1 | + (0.199 − 0.613i)2-s + (0.277 + 0.960i)3-s + (0.472 + 0.343i)4-s + (−0.966 + 0.313i)5-s + (0.644 + 0.0213i)6-s + (−0.952 + 1.31i)7-s + (0.826 − 0.600i)8-s + (−0.846 + 0.533i)9-s + 0.655i·10-s + (0.925 − 0.378i)11-s + (−0.198 + 0.549i)12-s + (1.29 + 0.419i)13-s + (0.614 + 0.845i)14-s + (−0.569 − 0.841i)15-s + (−0.0229 − 0.0704i)16-s + (−0.228 − 0.701i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.290 - 0.956i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.290 - 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.35151 + 1.00181i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.35151 + 1.00181i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-4.32 - 14.9i)T \) |
| 11 | \( 1 + (-371. + 151. i)T \) |
| good | 2 | \( 1 + (-1.12 + 3.46i)T + (-25.8 - 18.8i)T^{2} \) |
| 5 | \( 1 + (54.0 - 17.5i)T + (2.52e3 - 1.83e3i)T^{2} \) |
| 7 | \( 1 + (123. - 170. i)T + (-5.19e3 - 1.59e4i)T^{2} \) |
| 13 | \( 1 + (-787. - 255. i)T + (3.00e5 + 2.18e5i)T^{2} \) |
| 17 | \( 1 + (271. + 836. i)T + (-1.14e6 + 8.34e5i)T^{2} \) |
| 19 | \( 1 + (-68.7 - 94.5i)T + (-7.65e5 + 2.35e6i)T^{2} \) |
| 23 | \( 1 - 181. iT - 6.43e6T^{2} \) |
| 29 | \( 1 + (-2.73e3 - 1.98e3i)T + (6.33e6 + 1.95e7i)T^{2} \) |
| 31 | \( 1 + (-3.13e3 + 9.63e3i)T + (-2.31e7 - 1.68e7i)T^{2} \) |
| 37 | \( 1 + (-7.06e3 - 5.13e3i)T + (2.14e7 + 6.59e7i)T^{2} \) |
| 41 | \( 1 + (9.18e3 - 6.67e3i)T + (3.58e7 - 1.10e8i)T^{2} \) |
| 43 | \( 1 - 1.32e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 + (5.65e3 + 7.78e3i)T + (-7.08e7 + 2.18e8i)T^{2} \) |
| 53 | \( 1 + (-1.81e4 - 5.88e3i)T + (3.38e8 + 2.45e8i)T^{2} \) |
| 59 | \( 1 + (-1.18e4 + 1.63e4i)T + (-2.20e8 - 6.79e8i)T^{2} \) |
| 61 | \( 1 + (-1.69e4 + 5.50e3i)T + (6.83e8 - 4.96e8i)T^{2} \) |
| 67 | \( 1 - 2.18e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + (4.97e4 - 1.61e4i)T + (1.45e9 - 1.06e9i)T^{2} \) |
| 73 | \( 1 + (-2.26e3 + 3.12e3i)T + (-6.40e8 - 1.97e9i)T^{2} \) |
| 79 | \( 1 + (-1.72e4 - 5.60e3i)T + (2.48e9 + 1.80e9i)T^{2} \) |
| 83 | \( 1 + (4.03e3 + 1.24e4i)T + (-3.18e9 + 2.31e9i)T^{2} \) |
| 89 | \( 1 - 2.96e4iT - 5.58e9T^{2} \) |
| 97 | \( 1 + (1.76e3 - 5.44e3i)T + (-6.94e9 - 5.04e9i)T^{2} \) |
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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
−15.89904852677666953614081293286, −15.07898195249390517684381425836, −13.38003555842646224127910721161, −11.72960414000133036178979353973, −11.36727525608819550765270297485, −9.638789303005096961126179709659, −8.376488607339591931746220402233, −6.35590852449753357160177094186, −3.94763934394871909193091774970, −2.91492601891428558367189656468,
1.01307514881261496259201676264, 3.80982553775666754641009908463, 6.34177257872574465406573682591, 7.16598244792711068044361432966, 8.423796353217330255540280248207, 10.53690815931478530148578033278, 11.90427545103975580482628257931, 13.22508986493785865596562372542, 14.19322848919908336855655250447, 15.52687231336592621213252256061
