| L(s) = 1 | + (−0.427 − 1.31i)2-s + (−1.61 + 2.22i)3-s + (1.68 − 1.22i)4-s + (3.35 − 3.70i)5-s + (3.61 + 1.17i)6-s + (7.52 − 5.46i)7-s + (−6.81 − 4.95i)8-s + (0.449 + 1.38i)9-s + (−6.31 − 2.83i)10-s + (−5.23 + 9.67i)11-s + 5.71i·12-s + (−1.00 − 3.09i)13-s + (−10.4 − 7.56i)14-s + (2.81 + 13.4i)15-s + (−1.03 + 3.17i)16-s + (−2.14 + 6.59i)17-s + ⋯ |
| L(s) = 1 | + (−0.213 − 0.658i)2-s + (−0.538 + 0.740i)3-s + (0.421 − 0.305i)4-s + (0.671 − 0.740i)5-s + (0.603 + 0.195i)6-s + (1.07 − 0.780i)7-s + (−0.851 − 0.618i)8-s + (0.0499 + 0.153i)9-s + (−0.631 − 0.283i)10-s + (−0.475 + 0.879i)11-s + 0.476i·12-s + (−0.0772 − 0.237i)13-s + (−0.743 − 0.540i)14-s + (0.187 + 0.896i)15-s + (−0.0645 + 0.198i)16-s + (−0.125 + 0.387i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.624 + 0.780i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.624 + 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.01294 - 0.486905i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.01294 - 0.486905i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-3.35 + 3.70i)T \) |
| 11 | \( 1 + (5.23 - 9.67i)T \) |
| good | 2 | \( 1 + (0.427 + 1.31i)T + (-3.23 + 2.35i)T^{2} \) |
| 3 | \( 1 + (1.61 - 2.22i)T + (-2.78 - 8.55i)T^{2} \) |
| 7 | \( 1 + (-7.52 + 5.46i)T + (15.1 - 46.6i)T^{2} \) |
| 13 | \( 1 + (1.00 + 3.09i)T + (-136. + 99.3i)T^{2} \) |
| 17 | \( 1 + (2.14 - 6.59i)T + (-233. - 169. i)T^{2} \) |
| 19 | \( 1 + (3.57 - 4.91i)T + (-111. - 343. i)T^{2} \) |
| 23 | \( 1 - 34.5iT - 529T^{2} \) |
| 29 | \( 1 + (22.4 + 30.9i)T + (-259. + 799. i)T^{2} \) |
| 31 | \( 1 + (-13.9 - 43.0i)T + (-777. + 564. i)T^{2} \) |
| 37 | \( 1 + (-7.93 - 10.9i)T + (-423. + 1.30e3i)T^{2} \) |
| 41 | \( 1 + (-18.1 + 24.9i)T + (-519. - 1.59e3i)T^{2} \) |
| 43 | \( 1 - 60.1T + 1.84e3T^{2} \) |
| 47 | \( 1 + (26.4 - 36.3i)T + (-682. - 2.10e3i)T^{2} \) |
| 53 | \( 1 + (-13.3 + 4.32i)T + (2.27e3 - 1.65e3i)T^{2} \) |
| 59 | \( 1 + (40.2 - 29.2i)T + (1.07e3 - 3.31e3i)T^{2} \) |
| 61 | \( 1 + (24.3 + 7.92i)T + (3.01e3 + 2.18e3i)T^{2} \) |
| 67 | \( 1 + 80.0iT - 4.48e3T^{2} \) |
| 71 | \( 1 + (-19.9 + 61.4i)T + (-4.07e3 - 2.96e3i)T^{2} \) |
| 73 | \( 1 + (83.0 - 60.3i)T + (1.64e3 - 5.06e3i)T^{2} \) |
| 79 | \( 1 + (-35.7 + 11.6i)T + (5.04e3 - 3.66e3i)T^{2} \) |
| 83 | \( 1 + (-16.0 + 49.3i)T + (-5.57e3 - 4.04e3i)T^{2} \) |
| 89 | \( 1 - 33.3T + 7.92e3T^{2} \) |
| 97 | \( 1 + (107. - 34.8i)T + (7.61e3 - 5.53e3i)T^{2} \) |
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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
−15.09510743148127675521720278115, −13.69106932795473610292305499989, −12.38286421871620985668466731835, −11.12906934924869570632618522190, −10.37978384089716628100739392928, −9.520062056853410310018920665758, −7.66970313615907629726730131060, −5.67191598888311126174309590587, −4.49783975741120420779363702654, −1.70897403591820564611826317119,
2.43193474894523135173699382412, 5.62158527806598903973752800473, 6.51503049352165589579865916460, 7.69106001780238482936937806204, 8.958691004748051665253781623100, 10.96534996109965998292892657427, 11.67995035297801738800771509127, 12.91499767079223173575768450316, 14.39078522113563959078130363313, 15.15586847436401655827188444882
