| L(s) = 1 | + (−1.50 − 0.869i)2-s + (−1.65 − 2.49i)3-s + (−0.489 − 0.847i)4-s + (−1.93 − 1.11i)5-s + (0.325 + 5.20i)6-s + (2.93 − 6.35i)7-s + 8.65i·8-s + (−3.49 + 8.29i)9-s + (1.94 + 3.36i)10-s + (−11.0 + 6.38i)11-s + (−1.30 + 2.62i)12-s + 0.690·13-s + (−9.93 + 7.01i)14-s + (0.418 + 6.69i)15-s + (5.56 − 9.63i)16-s + (12.5 − 7.22i)17-s + ⋯ |
| L(s) = 1 | + (−0.752 − 0.434i)2-s + (−0.553 − 0.833i)3-s + (−0.122 − 0.211i)4-s + (−0.387 − 0.223i)5-s + (0.0542 + 0.867i)6-s + (0.419 − 0.907i)7-s + 1.08i·8-s + (−0.388 + 0.921i)9-s + (0.194 + 0.336i)10-s + (−1.00 + 0.580i)11-s + (−0.108 + 0.219i)12-s + 0.0530·13-s + (−0.709 + 0.501i)14-s + (0.0279 + 0.446i)15-s + (0.347 − 0.602i)16-s + (0.735 − 0.424i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.674 - 0.738i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.674 - 0.738i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.100699 + 0.228472i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.100699 + 0.228472i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.65 + 2.49i)T \) |
| 5 | \( 1 + (1.93 + 1.11i)T \) |
| 7 | \( 1 + (-2.93 + 6.35i)T \) |
| good | 2 | \( 1 + (1.50 + 0.869i)T + (2 + 3.46i)T^{2} \) |
| 11 | \( 1 + (11.0 - 6.38i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 - 0.690T + 169T^{2} \) |
| 17 | \( 1 + (-12.5 + 7.22i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (13.8 - 23.9i)T + (-180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (37.3 + 21.5i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + 26.6iT - 841T^{2} \) |
| 31 | \( 1 + (8.94 + 15.4i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (-17.5 + 30.4i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 - 15.0iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 23.1T + 1.84e3T^{2} \) |
| 47 | \( 1 + (38.1 + 22.0i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + (78.5 - 45.3i)T + (1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + (-21.3 + 12.3i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (33.0 - 57.3i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (13.6 + 23.5i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 + 76.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (-24.7 - 42.8i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-18.5 + 32.0i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + 20.0iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (-62.7 - 36.2i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 - 23.2T + 9.40e3T^{2} \) |
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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
−12.65392624496501072923997798723, −11.69058057034368634081958715502, −10.63040645153720850426689775643, −9.974375869975597860904010726483, −8.111458590422587627110033742912, −7.72747522293431053809155896375, −5.95923023660269113785734224539, −4.60101564147969526424409287546, −1.91323524993802085227931602854, −0.24914812453454057573550143046,
3.37409648568818890719972716321, 4.97888841736827729409867100212, 6.29030969510584420444650657345, 7.893373030137760301412346195289, 8.700799252424127342261064941147, 9.808748014978193955445427225397, 10.88276972594052094514713185406, 11.89418542460915147324162831622, 12.95987489508085182976771505675, 14.55143346744060212071918344157
